UNIVERSITY OF CALIFORNIA, SAN DIEGO
Structural Characterization of Concrete Filled
Fiber Reinforced Shells
A dissertation submitted in partial satisfaction of the
requirements for the degree Doctor of Philosophy in
Engineering Science
(Structural Engineering)
by
Andrew Davol
Committee in charge:
Professor Frieder Seible Professor Gilbert Hegemier Professor Vistasp Karbhari Professor Donald Smith Professor Audrey Terras
1998
Copyright
Andrew Davol, 1998
All rights reserved.
iii
The dissertation of Andrew Davol is approved, and it
is acceptable in quality and form for publication on
microfilm:
____________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________ Chair
University of California, San Diego
1998
iv
Dedication
This work is dedicated to my first true companion Max, whose boundless energy was a constant inspiration to me, and to my life companion Michele, whose vision of our
shared future has made the completion of this dissertation possible.
v
TABLE OF CONTENTS
SIGNATURE PAGE .................................................................................................... iii
DEDICATION...............................................................................................................iv
TABLE OF CONTENTS................................................................................................v
LIST OF FIGURES .................................................................................................... viii
LIST OF TABLES.................................................................................................... xxiii
LIST OF SYMBOLS ..................................................................................................xxv
ABSTRACT..............................................................................................................xxxii
1. INTRODUCTION..................................................................................................1
2. RESEARCH REVIEW..........................................................................................7
3. MATERIAL CHARACTERIZATION..............................................................13 3.1 ADVANCED COMPOSITE SHELLS.......................................................................13
3.1.1 Fiber Reinforcement .................................................................................13 3.1.2 Matrix Materials .......................................................................................15 3.1.3 Manufacturing Processes ..........................................................................17 3.1.4 Typical Ply Properties...............................................................................17 3.1.5 Classical Lamination Theory ....................................................................18 3.1.6 Equivalent Plate Properties .......................................................................27 3.1.7 First Ply Failure Criteria ...........................................................................28 3.1.8 Thermal Expansion ...................................................................................28
3.2 CONCRETE........................................................................................................31 3.2.1 Uniaxial Compression...............................................................................32 3.2.2 Biaxial and Triaxial Stress States - Confinement Effects.........................32 3.2.3 Tension......................................................................................................35
4. ANALYTICAL MODELING OF CONCRETE FILLED FRP SHELLS......38 4.1 CIRCULAR SHELLS............................................................................................38
4.1.1 Compression .............................................................................................38 4.1.2 Tension......................................................................................................50 4.1.3 Shear .........................................................................................................50 4.1.4 Bending.....................................................................................................54
4.2 CONREC SHELLS...............................................................................................56 4.2.1 Compression .............................................................................................57 4.2.2 Bending.....................................................................................................68
vi
5. EXPERIMENTAL PROGRAM TO VALIDATE CONCRETE FILLED FRP TUBE BEHAVIOR.....................................................................................................71 5.1 SMALL SCALE SHELLS......................................................................................71
5.1.1 Concrete Characterization.........................................................................74 5.1.2 Compression .............................................................................................76 5.1.3 Bending.....................................................................................................89
5.2 FULL SCALE BENDING TESTS .........................................................................110 5.2.1 Concrete Properties.................................................................................118 5.2.2 Hollow Shell ...........................................................................................119 5.2.3 Concrete Filled Shells.............................................................................123 5.2.4 Concrete Filled Shell with integral Concrete Deck ................................137
6. CORRELATION OF ANALYTICAL MODELS TO EXPERIMENTAL DATA .........................................................................................................................145 6.1 CIRCULAR SHELLS..........................................................................................145
6.1.1 Small Scale Shells...................................................................................145 6.1.2 Full Scale Specimens ..............................................................................153
6.2 CONREC SHELLS.............................................................................................159 6.2.1 Compression ...........................................................................................159 6.2.2 Bending...................................................................................................162
7. PARAMETER STUDIES OF MATERIAL, LAY-UP, THICKNESS AND SHAPE VARIATIONS.............................................................................................168 7.1 CIRCULAR SHELLS..........................................................................................169
7.1.1 Compression Behavior............................................................................169 7.1.2 Bending Behavior ...................................................................................172
7.2 CONREC SHELLS.............................................................................................184 7.3 HYBRID SHELLS..............................................................................................187
7.3.1 Compression ...........................................................................................188 7.3.2 Bending...................................................................................................188
8. STRESS CONCENTRATIONS, TENSION STIFFENING AND THERMAL EXPANSION EFFECTS..........................................................................................189 8.1 STRESS CONCENTRATIONS....................................................................189
8.1.1 Closed Form Solution .............................................................................189 8.1.2 Parameter Study......................................................................................192 8.1.3 Case Study ..............................................................................................198
8.2 EFFECTS OF TENSION STIFFENING ......................................................201 8.3 THERMAL EFFECTS ..................................................................................204
8.3.1 Thermal Strains in Circular Sections ......................................................204 8.3.2 Thermal Testing......................................................................................205 8.3.3 Parameter Studies for Thermally Induced Strains ..................................207
8.4 LOCAL COMPRESSION BUCKLING OF CONCRETE FILLED FRP SHELLS...........209
9. CONCLUSIONS ................................................................................................211
vii
APPENDIX - MOMENT CURVATUR PROGRAM............................................220
REFERENCES..........................................................................................................236
viii
LIST OF FIGURES
FIGURE 1-1 CONVENTIONALLY REINFORCED CONCRETE COLUMN ....5
FIGURE 1-2 CARBON FIBER WRAP APPLIED TO BRIDGE COLUMN FOR
SEISMIC RETROFIT ......................ERROR! BOOKMARK NOT DEFINED.
FIGURE 1-3 CONCEPT FOR CONCRETE FILLED FIBER REINFORCED
SHELL ....................................................................................................................6
FIGURE 1-4 PROTOTYPE BRIDGE STRUCTURE WITH CARBON SHELL
GIRDERS AND A FIBER GLASS DECK SYSTEM.........................................6
FIGURE 2-1 AREA AND VOLUME STRAIN DEFINITION ..............................12
FIGURE 2-2 EXPANSION BEHAVIOR OF CONCRETE ...................................12
FIGURE 3-1 MATERIAL AND STRUCTURAL COORDINATE SYSTEMS ...23
FIGURE 3-2 GEOMETRY OF LAMINATE ..........................................................25
FIGURE 3-3 STRESS STRAIN MODELS FOR CONFINED CONCRETE.......34
FIGURE 4-1 EXPERIMENTAL AND SMOOTHED DILATION RATE ...........42
FIGURE 4-2 CONCRETE TANGENT MODULUS VS. RADIAL STRAIN.......43
FIGURE 4-3 EQUIVALENT TANGENT POISSON'S RATIO FOR TEST
CYLINDERS ........................................................................................................46
FIGURE 4-4 ANALYTICAL EQUIVALENT TANGENT POISSON'S
RATIO...................................................................................................................46
FIGURE 4-5 MAXIMUM TANGENT POISSON'S RATIO VS. HYDROSTATIC
PRESSURE ...........................................................................................................47
FIGURE 4-6 ANALYTICAL MODEL FOR COMPRESSION BEHAVIOR......48
ix
FIGURE 4-7 DETERMINATION OF MAXIMUM EQUIVALENT TANGENT
POISSON'S RATIO.............................................................................................49
FIGURE 4-8 SHEAR TRANSFER BETWEEN CONCRETE CORE AND
COMPOSITE SHELL .........................................................................................51
FIGURE 4-9 GEOMETRIC PROPERTIES FOR DETERMINATION OF
SHEAR STRESS ..................................................................................................53
FIGURE 4-10 ANALYSIS FLOW FOR BENDING BEHAVIOR ........................54
FIGURE 4-11 CONREC CROSS SECTION ...........................................................57
FIGURE 4-12 FINITE ELEMENT MODEL USED FOR EVALUATION OF
CONREC SECTIONS .........................................................................................59
FIGURE 4-13 CONREC GEOMETRIES USED FOR THIS ANALYSIS ...........61
FIGURE 4-14 AREA STRAIN RATIO PROFILE, 0% ±10O PLIES, FLAT TO
RADIUS RATIO .5 ..............................................................................................62
FIGURE 4-15 AREA STRAIN RATIO PROFILE, 0% ±10O FIBERS, FLAT TO
RADIUS RATIO OF 1.........................................................................................62
FIGURE 4-16 AREA STRAIN RATIO PROFILE, 0% ±10O FIBERS, FLAT TO
RADIUS RATIO OF 2.........................................................................................63
FIGURE 4-17 AREA STRAIN RATIO PROFILE, 0% ±10O FIBERS, FLAT TO
RADIUS RATIO OF 3.........................................................................................63
FIGURE 4-18 AREA STRAIN RATIO FOR 0% ±10O CONREC SHELL..........64
FIGURE 4-19 AREA STRAIN RATIO FOR 50% ±10O CONREC SHELL........64
FIGURE 4-20 AREA STRAIN RATIO FOR 80% ±10O CONREC SHELL........65
x
FIGURE 4-21 HOOP STRESS IN SHELLS WITH FLAT TO RADIUS RATIO
OF .5 ......................................................................................................................66
FIGURE 4-22 HOOP STRESS IN SHELLS WITH FLAT TO RADIUS RATIO
OF 1 .......................................................................................................................66
FIGURE 4-23 HOOP STRESS IN SHELLS WITH FLAT TO RADIUS RATIO
OF 2 .......................................................................................................................67
FIGURE 4-24 HOOP STRESS IN SHELLS WITH FLAT TO RADIUS RATIO
OF 3 .......................................................................................................................67
FIGURE 4-25 MOMENT CURVATURE OF TYPICAL CONREC
SECTION..............................................................................................................69
FIGURE 4-26 CONCRETE STRESS STRAIN RELATION.................................70
FIGURE 4-27 COMPARATIVE MOMENT CURVATURE FOR CONREC
SECTION WITH VARIOUS CONCRETE MODELS ....................................70
FIGURE 5-1 NOMINAL GEOMETRY OF CONREC SECTION .......................72
FIGURE 5-2 CONCRETE COMPRESSION STRESS STRAIN RELATION....75
FIGURE 5-3 HOOP STRAIN VS. LONGITUDINAL STRAIN FOR
CONCRETE CYLINDERS UNDER UNIAXIAL COMPRESSION .............76
FIGURE 5-4 COMPARISON OF HOOP STRAINS FOR CYLINDERS WITH
VARIOUS ASPECT RATIOS ............................................................................77
FIGURE 5-5 TYPICAL COMPRESSION TEST SETUP......................................78
FIGURE 5-6 STRAIN GAGE LAYOUT FOR CIRCULAR COMPRESSION
SPECIMENS ........................................................................................................82
xi
FIGURE 5-7 LOAD VS. STRAIN CURVES FOR CIRCULAR CYLINDERS...82
FIGURE 5-8 HOOP VS. LONGITUDINAL STRAINS FOR CIRCULAR
CYLINDERS ........................................................................................................83
FIGURE 5-9 LONGITUDINAL STRAIN IN HELICAL CIRCULAR
CYLINDERS ........................................................................................................83
FIGURE 5-10 CONCRETE STRESS STRAIN CURVES FOR ALL HOOP
CIRCULAR CYLINDERS..................................................................................84
FIGURE 5-11 TYPICAL FAILURE OF ALL HOOP CIRCULAR SHELL .......84
FIGURE 5-12 FAILURE OF HELICAL CIRCULAR SHELLS ..........................85
FIGURE 5-13 STRAIN GAGE LAYOUT FOR CONREC COMPRESSION
SPECIMENS ........................................................................................................86
FIGURE 5-14 LOAD VS. LONGITUDINAL STRAIN CONREC
CYLINDERS ........................................................................................................87
FIGURE 5-15 HOOP VS. LONGITUDINAL STRAIN FOR CONREC
CYLINDERS ........................................................................................................87
FIGURE 5-16 TYPICAL FAILURE OF ALL HOOP CONREC SHELL ...........88
FIGURE 5-17 FAILURE OF CONREC HELICAL SHELLS...............................88
FIGURE 5-18 SCHEMATIC OF FOUR POINT BENDING TEST SETUP........90
FIGURE 5-19 FOUR POINT BENDING TEST ON SMALL SCALE
SPECIMEN...........................................................................................................90
FIGURE 5-20 STRAIN GAGE LAYOUT FOR CIRCULAR BENDING
SPECIMENS ........................................................................................................91
xii
FIGURE 5-21 LOAD - DISPLACEMENT CURVE FOR THIN CIRCULAR
BENDING SPECIMEN .......................................................................................93
FIGURE 5-22 STRAIN PROFILE FOR THIN CIRCULAR SECTION IN
CONSTANT MOMENT REGION ....................................................................93
FIGURE 5-23 STRAIN PROFILE FOR THIN CIRCULAR SECTION IN
SHEAR AREA......................................................................................................94
FIGURE 5-24 LONGITUDINAL STRAIN VS. MOMENT FOR THIN
CIRCULAR SPECIMEN ....................................................................................94
FIGURE 5-25 HOOP STRAIN VS. MOMENT FOR THIN CIRCULAR
SPECIMEN...........................................................................................................95
FIGURE 5-26 SHEAR STRAIN VS. SHEAR FOR THIN CIRCULAR
SPECIMEN...........................................................................................................95
FIGURE 5-27 LOAD - DISPLACEMENT CURVE FOR THICK CIRCULAR
BENDING SPECIMEN .......................................................................................97
FIGURE 5-28 STRAIN PROFILE FOR THICK CIRCULAR SECTION IN
CONSTANT MOMENT REGION ....................................................................97
FIGURE 5-29 STRAIN PROFILE FOR THICK CIRCULAR SECTION IN
SHEAR SPAN.......................................................................................................98
FIGURE 5-30 LONGITUDINAL STRAIN VS. MOMENT FOR THICK
CIRCULAR SECTION .......................................................................................98
FIGURE 5-31 HOOP STRAIN VS. MOMENT FOR THICK CIRCULAR
SECTION..............................................................................................................99
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FIGURE 5-32 SHEAR STRAIN VS. APPLIED SHEAR FOR THICK
CIRCULAR SPECIMEN ....................................................................................99
FIGURE 5-33 FAILURE OF THICK CIRCULAR SECTION ...........................100
FIGURE 5-34 STRAIN GAGE LAYOUT FOR CONREC BENDING
SPECIMENS ......................................................................................................101
FIGURE 5-35 LOAD - DISPLACEMENT CURVE FOR THIN CONREC
BENDING SPECIMEN .....................................................................................102
FIGURE 5-36 STRAIN PROFILE FOR THIN CONREC SECTION IN
CONSTANT MOMENT REGION ..................................................................102
FIGURE 5-37 STRAIN PROFILE FOR THIN CONREC SECTION IN SHEAR
AREA ..................................................................................................................103
FIGURE 5-38 LONGITUDINAL STRAIN VS. MOMENT FOR THIN CONREC
SPECIMEN.........................................................................................................103
FIGURE 5-39 HOOP STRAIN VS. MOMENT FOR THIN CONREC
SPECIMEN.........................................................................................................104
FIGURE 5-40 SHEAR STRAIN VS. APPLIED SHEAR FOR THIN CONREC
SPECIMEN.........................................................................................................104
FIGURE 5-41 FAILURE OF THIN CONREC SECTION ..................................105
FIGURE 5-42 LOAD-DISPLACEMENT RESPONSE FOR THICK CONREC
SPECIMEN.........................................................................................................106
FIGURE 5-43 STRAIN PROFILE FOR THICK CONREC SECTION IN
CONSTANT MOMENT REGION ..................................................................107
xiv
FIGURE 5-44 STRAIN PROFILE FOR THICK CONREC SECTION IN
SHEAR AREA....................................................................................................107
FIGURE 5-45 LONGITUDINAL STRAIN VS. MOMENT FOR THICK
CONREC ............................................................................................................108
FIGURE 5-46 HOOP STRAIN VS. MOMENT FOR THICK CONREC
SECTION............................................................................................................108
FIGURE 5-47 SHEAR STRAIN VS. APPLIED SHEAR FOR THICK CONREC
SPECIMEN.........................................................................................................109
FIGURE 5-48 FAILURE OF THICK CONREC SECTION ...............................109
FIGURE 5-49 FULL SCALE FOUR POINT BENDING TEST..........................110
FIGURE 5-50 SCHEMATIC OF FULL SCALE BENDING TESTS .................111
FIGURE 5-51 ENDBLOCK FOR SUPPORT OF BENDING TEST
SPECIMENS ......................................................................................................112
FIGURE 5-52 SHELL END DIAPHRAGM ..........................................................113
FIGURE 5-53 STEEL CONNECTION CAGE......................................................114
FIGURE 5-54 SHELL WITH CONNECTING CAGE .........................................114
FIGURE 5-55 ENDBLOCK LOWER SECTION .................................................115
FIGURE 5-56 PLACEMENT OF SHELL INTO ENDBLOCKS........................115
FIGURE 5-57 PVC PIPE FOR PUMPING SHELL .............................................116
FIGURE 5-58 COMPLETED END BLOCK FORM............................................116
FIGURE 5-59 INSTRUMENTATION LAYOUT FOR HOLLOW SHELL
TEST....................................................................................................................121
xv
FIGURE 5-60 LOAD DISPLACEMENT CURVE FOR HOLLOW SHELL
TESTS .................................................................................................................121
FIGURE 5-61 LONGITUDINAL AND HOOP STRAINS FOR HOLLOW
SHELLS ..............................................................................................................122
FIGURE 5-62 SHEAR STRAINS FOR HOLLOW SHELL TESTS ..................122
FIGURE 5-63 DISPLACEMENT INSTRUMENTATION FOR FILLED SHELL
TEST....................................................................................................................123
FIGURE 5-64 STRAIN GAGE LOCATIONS AND DESIGNATION FOR
FILLED TUBE TESTS......................................................................................124
FIGURE 5-65 LOAD DISPLACEMENT PLOT FOR FILLED SHELL
TEST #1...............................................................................................................126
FIGURE 5-66 STRAIN PROFILE IN CONSTANT MOMENT SECTION,
FILLED SHELL TEST #1 ................................................................................127
FIGURE 5-67 STRAIN PROFILE IN SHEAR SECTION, FILLED SHELL
TEST #1...............................................................................................................127
FIGURE 5-68 LONGITUDINAL STRAIN VS. MOMENT IN COMP. ZONE
FOR FILLED SHELL #1 ..................................................................................128
FIGURE 5-69 LONGITUDINAL STRAIN VS. MOMENT IN TENSION ZONE
FOR FILLED SHELL #1 ..................................................................................128
FIGURE 5-70 HOOP STRAIN VS. MOMENT IN COMPRESSION ZONE FOR
FILLED SHELL #1............................................................................................129
xvi
FIGURE 5-71 HOOP STRAIN VS. MOMENT IN TENSION ZONE FOR
FILLED SHELL #1............................................................................................129
FIGURE 5-72 SHEAR STRAIN VS. APPLIED SHEAR FOR FILLED
SHELL #1 ...........................................................................................................130
FIGURE 5-73 FAILURE OF FILLED SHELL #1................................................130
FIGURE 5-74 LOAD DISPLACEMENT PLOT FOR FILLED SHELL
TEST #2...............................................................................................................132
FIGURE 5-75 STRAIN PROFILE IN CONSTANT MOMENT SECTION,
FILLED SHELL TEST #2 ................................................................................133
FIGURE 5-76 STRAIN PROFILE IN SHEAR SECTION, FILLED SHELL
TEST #2...............................................................................................................133
FIGURE 5-77 LONGITUDINAL STRAIN VS. MOMENT IN COMP. ZONE
FOR FILLED SHELL #2 ..................................................................................134
FIGURE 5-78 LONGITUDINAL STRAIN VS. MOMENT IN TENSION ZONE
FOR FILLED SHELL #2 ..................................................................................134
FIGURE 5-79 HOOP STRAIN VS. MOMENT IN COMPRESSION ZONE FOR
FILLED SHELL #2............................................................................................135
FIGURE 5-80 HOOP STRAIN VS. MOMENT IN TENSION ZONE FOR
FILLED SHELL #2............................................................................................135
FIGURE 5-81 SHEAR STRAIN VS. APPLIED SHEAR FOR FILLED
SHELL #2 ...........................................................................................................136
FIGURE 5-82 FAILURE OF FILLED SHELL #2................................................136
xvii
FIGURE 5-83 DISPLACEMENT INSTRUMENTATION FOR FILLED SHELL
WITH SLAB TEST............................................................................................138
FIGURE 5-84 STRAIN GAGE LOCATIONS AND DESIGNATION FOR
SHELL ................................................................................................................138
FIGURE 5-85 STRAIN GAGES PLACED ON SHEAR CONNECTION
DOWELS ............................................................................................................139
FIGURE 5-86 INSTRUMENTATION LOCATIONS AND DESIGNATION FOR
TOP MAT OF STEEL REINFORCEMENT IN DECK................................139
FIGURE 5-87 STRESS CONCENTRATION STRAIN GAGE LOCATIONS
AND DESIGNATION........................................................................................140
FIGURE 5-88 LOAD DISPLACEMENT ENVELOPE FOR FILLED SHELL
WITH SLAB .......................................................................................................142
FIGURE 5-89 STRAIN PROFILE ACROSS SECTION FOR FILLED SHELL
WITH SLAB .......................................................................................................142
FIGURE 5-90 STRESS CONCENTRATION AROUND PENETRATION IN
SHEAR SPAN.....................................................................................................143
FIGURE 5-91 STRESS CONCENTRATION IN THE CONST. MOMENT
REGION..............................................................................................................143
FIGURE 6-1 CONCRETE STRESS VS. STRAIN FOR SMALL SCALE
COMPRESSION SPECIMENS........................................................................146
FIGURE 6-2 RADIAL VS. LONGITUDINAL STRAIN FOR SMALL SCALE
COMP. SPECIMENS ........................................................................................147
xviii
FIGURE 6-3 LOAD VS. DISPLACEMENT FOR SMALL SCALE CIRCULAR
SECTIONS..........................................................................................................150
FIGURE 6-4 STRAINS IN CONSTANT MOMENT REGION FOR THIN
CIRCULAR SHELL ..........................................................................................150
FIGURE 6-5 STRAINS IN CONSTANT MOMENT REGION FOR THICK
CIRCULAR SHELL ..........................................................................................151
FIGURE 6-6 STRAINS IN SHEAR AREA FOR THIN CIRCULAR
SHELL ................................................................................................................151
FIGURE 6-7 STRAINS IN SHEAR SPAN FOR THICK CIRCULAR
SHELL ................................................................................................................152
FIGURE 6-8 SHEAR STRAIN IN THIN CIRCULAR SHELL ..........................152
FIGURE 6-9 SHEAR STRAIN IN THICK CIRCULAR SHELL .......................153
FIGURE 6-10 LOAD DISPLACEMENT CURVES FOR FULL SCALE FILLED
SHELL TESTS ...................................................................................................155
FIGURE 6-11 EXTREME FIBER STRAINS IN CONSTANT MOMENT
REGION FOR SHELL #1.................................................................................156
FIGURE 6-12 EXTREME FIBER STRAINS IN SHEAR SPAN FOR
SHELL #1 ...........................................................................................................156
FIGURE 6-13 EXTREME FIBER STRAINS IN CONSTANT MOMENT
REGION FOR SHELL #2.................................................................................157
FIGURE 6-14 EXTREME FIBER STRAINS IN SHEAR SPAN FOR
SHELL #2 ...........................................................................................................157
xix
FIGURE 6-15 SHELL CENTERLINE SHEAR STRAINS FOR SHELL #1 .....158
FIGURE 6-16 SHELL CENTERLINE SHEAR STRAINS FOR SHELL #2 .....158
FIGURE 6-17 LOAD VS. LONGITUDINAL STRAIN FOR THICK CONREC
CYLINDERS ......................................................................................................161
FIGURE 6-18 HOOP STRAINS VS. LONGITUDINAL STRAIN IN THICK
CONREC SECTION .........................................................................................161
FIGURE 6-19 LOAD DISPLACEMENT FOR CONREC BENDING
SPECIMENS ......................................................................................................164
FIGURE 6-20 STRAINS IN CONSTANT MOMENT REGION FOR THIN
CONREC SHELL ..............................................................................................164
FIGURE 6-21 STRAINS IN CONSTANT MOMENT REGION FOR THICK
CONREC SHELL ..............................................................................................165
FIGURE 6-22 STRAINS IN SHEAR AREA FOR THIN CONREC SHELL ....165
FIGURE 6-23 STRAINS IN SHEAR AREA FOR THICK CONREC
SHELL ................................................................................................................166
FIGURE 6-24 SHEAR STRAIN IN THIN CONREC SHELL ............................166
FIGURE 6-25 SHEAR STRAIN IN THICK CONREC SHELL .........................167
FIGURE 7-1 COMPRESSION BEHAVIOR OF CARBON EPOXY
SHELLS ..............................................................................................................171
FIGURE 7-2 COMPRESSION BEHAVIOR OF E-GLASS SHELLS................171
FIGURE 7-3 CONFINEMENT EFFICIENCY OF E-GLASS VS. CARBON
SHELLS ..............................................................................................................172
xx
FIGURE 7-4 GEOMETRY FOR MOMENT CALCULATION .........................173
FIGURE 7-5 NORMALIZED MOMENT CURVATURE, R/T=10, CARBON
EPOXY SHELL .................................................................................................175
FIGURE 7-6 NORMALIZED MOMENT CURVATURE, R/T=15, CARBON
EPOXY SHELL .................................................................................................175
FIGURE 7-7 NORMALIZED MOMENT CURVATURE, R/T=20, CARBON
EPOXY SHELL .................................................................................................176
FIGURE 7-8 NORMALIZED MOMENT CURVATURE, R/T=25, CARBON
EPOXY SHELL .................................................................................................176
FIGURE 7-9 NORMALIZED MOMENT CURVATURE, R/T=10, E-GLASS
SHELL ................................................................................................................177
FIGURE 7-10 NORMALIZED MOMENT CURVATURE, R/T=15, E-GLASS
SHELL ................................................................................................................177
FIGURE 7-11 NORMALIZED MOMENT CURVATURE, R/T=20, E-GLASS
SHELL ................................................................................................................178
FIGURE 7-12 NORMALIZED MOMENT CURVATURE, R/T=25, E-GLASS
SHELL ................................................................................................................178
FIGURE 7-13 MOMENT CURVATURE WITH AXIAL LOAD, R/T=10, 10%
HELICAL FIBERS ............................................................................................180
FIGURE 7-14 MOMENT CURVATURE WITH AXIAL LOAD, R/T=10, 50%
HELICAL FIBERS ............................................................................................180
xxi
FIGURE 7-15 MOMENT CURVATURE WITH AXIAL LOAD, R/T=10, 90%
HELICAL FIBERS ............................................................................................181
FIGURE 7-16 MOMENT CURVATURE WITH AXIAL LOAD, R/T=25, 10%
HELICAL FIBERS ............................................................................................181
FIGURE 7-17 MOMENT CURVATURE WITH AXIAL LOAD, R/T=25, 50%
HELICAL FIBERS ............................................................................................182
FIGURE 7-18 MOMENT CURVATURE WITH AXIAL LOAD, R/T=25, 90%
HELICAL FIBERS ............................................................................................182
FIGURE 7-19 FLEXURAL STIFFNESS OF E-GLASS VS. CARBON
SHELLS ..............................................................................................................183
FIGURE 7-20 MOMENT CURVATURE RESPONSE FOR CONREC
SECTIONS, D/T=20...........................................................................................186
FIGURE 7-21 MOMENT CURVATURE RESPONSE FOR CONREC
SECTIONS, D/T=50...........................................................................................186
FIGURE 7-22 NORMALIZED MOMENT CURVATURE RESPONSE FOR
CONREC SECTIONS, D/T=20 ........................................................................187
FIGURE 8-1 INFINITE PLATE WITH A CIRCULAR INCLUSION...............191
FIGURE 8-2 LOAD CASES FOR STRESS CONCENTRATION STUDY .......194
FIGURE 8-3 STRESS CONCENTRATION FACTORS......................................197
FIGURE 8-4 TANGENTIAL STRESS CONCENTRATION VARIATION
AROUND HOLE FOR 80% HELICAL SHELL ...........................................197
FIGURE 8-5 SHEAR CONNECTION ...................................................................198
xxii
FIGURE 8-6 FINITE ELEMENT MODEL FOR STRESS CONCENTRATION
STUDIES.............................................................................................................201
FIGURE 8-7 AVERAGE STRESS VS. AVERAGE STRAIN FOR TENSION
STIFFENING .....................................................................................................203
FIGURE 8-8 LOAD IN CONCRETE FILLED CARBON SHELL IN PURE
TENSION WITH AND WITHOUT TENSION STIFFENING
EFFECTS............................................................................................................203
FIGURE 8-9 THERMALLY INDUCED MECHANICAL STRAINS PER
DEGREE CENTIGRADE.................................................................................206
FIGURE 8-10 COEFFICIENTS OF THERMAL EXPANSION FOR FIBER
REINFORCED SHELLS ..................................................................................208
FIGURE 8-11 HOOP STRESS IN SHELL 90O PLIES DUE TO A
TEMPERATURE RISE OF 55 OC...................................................................208
FIGURE 8-12 ULTIMATE BUCKLING STRAIN FOR ALL BENDING
SPECIMENS ......................................................................................................210
xxiii
LIST OF TABLES
TABLE 3-1 TYPICAL PROPERTIES OF COMMERCIAL GLASS FIBER
REINFORCEMENTS ........................................................................................14
TABLE 3-2 MECHANICAL PROPERTIES FOR SELECT CARBON
FIBERS ................................................................................................................15
TABLE 3-3 MECHANICAL PROPERTIES FOR COMMON
THERMOSETTING RESINS ...........................................................................16
TABLE 3-4 TYPICAL PLY PROPERTIES FOR FIBER-REINFORCED
EPOXY RESINS ..................................................................................................19
TABLE 4-1 CONSTANTS FOR TANGENT MODULUS RELATION ...............43
TABLE 4-2 COMPOSITE LAY-UPS USED FOR CONREC STUDIES .............61
TABLE 5-1 SMALL SCALE TEST SHELLS .........................................................73
TABLE 5-2 VENDOR SUPPLIED PLY PROPERTIES ........................................73
TABLE 5-3 EQUIVALENT PLATE PROPERTIES..............................................74
TABLE 5-4 EXPERIMENTALLY DERIVED CONCRETE PROPERTIES......76
TABLE 5-5 COMPRESSION SPECIMENS ...........................................................79
TABLE 5-6 SHELLS FOR SMALL SCALE BENDING TESTS ..........................89
TABLE 5-7 COMPOSITE ARCHITECTURES FOR LARGE SCALE
TESTS .................................................................................................................117
TABLE 5-8 VENDOR SUPPLIED PLY PROPERTIES ......................................117
TABLE 5-9 EQUIVALENT PLATE PROPERTIES FOR LARGE SCALE
TESTS .................................................................................................................118
xxiv
TABLE 5-10 CONCRETE MIX USED FOR FILLED SHELLS ........................119
TABLE 5-11 CONCRETE PROPERTIES FOR FILLED SHELL TESTS .......119
TABLE 7-1 PLY PROPERTIES FOR PARAMETER STUDIES.......................169
TABLE 7-2 GEOMETRY OF CONREC SECTIONS FOR NORMALIZED
COMPARISON ..................................................................................................185
TABLE 8-1 COMPOSITE ARCHITECTURES FOR STRESS
CONCENTRATION STUDY ...........................................................................195
TABLE 8-2 STRESS CONCENTRATION FACTORS........................................196
TABLE 8-3 STRESS AROUND PENETRATION FOR BEAM AND SLAB
SHEAR CONNECTION ...................................................................................200
TABLE 8-4 PLY PROPERTIES FOR THERMAL TESTING ...........................206
xxv
LIST OF SYMBOLS
Scalars
Ac � cross sectional area of concrete core
E1 � ply modulus in fiber direction
E2 � ply modulus transverse to fiber direction
Ec � concrete tangent modulus
Eca � average tangent stiffness of concrete core
Eco � initial concrete modulus
EH � equivalent modulus in hoop direction for shell
EL � equivalent modulus in longitudinal direction for shell
Ex � modulus in x direction
Ey � modulus in y direction
f - flat length for conrec section
f�c � compression strength of unconfined concrete
fc � concrete stress
ft � tension strength of concrete
G12 � ply in-plane shear modulus
Gxy � shear modulus in x-y plane
hn � distance from mid-plane to near surface of ply n
M � moment in section
Mx � moment applied to laminate about y axis
MxT � equivalent thermal moment about y axis
xxvi
Mxy � twisting moment applied to laminate
MxyT � equivalent thermal twisting moment
My � moment applied to laminate about x axis
MyT � equivalent thermal moment about x axis
Nx � force applied to laminate in x direction
NxT � equivalent thermal force on laminate in x direction
Nxy � in-plane shear force applied to laminate
NxyT � equivalent thermal in-plane shear force applied to laminate
Ny � force applied to laminate in y direction
NyT � equivalent thermal force applied to laminate in y direction
Pc � load in concrete
Ps � load in shell
qc � shear flow between shell and core
qs � shear flow in shell
r - radius for conrec section
R � shell mean radius
t � thickness of laminated shell
teff � effective thickness of concrete used for shear calculation
u � displacement in x direction
uo � mid-plane displacement in x direction
v � displacement in y direction
V � shear load on section
xxvii
Vc � shear capacity of concrete core
vo � mid-plane displacement in y direction
w � displacement in z direction
wo � mid-plane displacement in z direction
αx � coefficient of thermal expansion in structural x direction
αxy � apparent shear coefficient of thermal expansion in structural x-y plane
αy � coefficient of thermal expansion in structural y direction
α1 − ply coefficient of thermal expansion in fiber direction
α2 − ply coefficient of thermal expansion normal to fiber direction
β1 − bond characteristic factor for tension stiffening
β2 − loading characteristic factor for tension stiffening
ε1 � ply strain in fiber direction or longitudinal concrete strain
εa � area strain
εcf � average strain for tension stiffening
εv � volume strain
εx � strain in structural x direction
εxo � mid-plane strain in structural x direction
εxT � thermal strain in structural x direction
εy � strain in structural y direction
εyo � mid-plane strain in structural y direction
εyT � thermal strain in structural y direction
xxviii
ε1Τ − ply thermal strain in fiber direction
ε2 − ply strain normal to fiber direction
ε2Τ − ply thermal strains normal to fiber direction
ε3 − ply strain normal to fiber direction
γxy � shear strain in structural x-y plane
γxyT � thermal shear strain in structural x-y plane
γ12 − ply shear strain 1-2 plane
γ13 − ply shear strain 1-3 plane
γ23 − ply shear strain 2-3 plane
κx � curvature about y axis
κxy � twist
κy � curvature about x axis
λ − concrete density factor
ν12 − ply in-plane Poisson�s ratio for loading in the fiber direction
ν21 − ply in-plane Poisson�s ratio for loading normal to the fiber direction
θ − rotation angle between ply and structure coordinate systems or angle around hole
for stress concentration analysis
σx � stress in structural x direction
σy �stress in structural y direction
σz � stress in structural z direction
σ1 − ply stress in fiber direction or longitudinal concrete stress
xxix
σ2 − ply stress normal to fiber direction
σ3 − ply stress normal to fiber direction
τxy � shear stress in x-y plane
τ12 − ply shear stress 1-2 plane
τ13 − ply shear stress 1-3 plane
τ23 − ply shear stress 2-3 plane
εL � longitudinal strain in shell
εH � hoop strain in shell
νLH � Poisson�s ratio in shell for loading in the longitudinal direction
νHL � Poisson�s ratio in shell for loading in the hoop direction
νc � concrete equivalent tangent Poisson�s ratio
νco � initial concrete Poisson�s ratio
σL � longitudinal stress in shell
σH � hoop stress in shell
εr � radial strain in concrete core
σr � radial stress in concrete core
µ − dilation rate
T − temperature
σTx - thermal stress in x direction
σTy - thermal strain in y direction
τTxy
- thermal shear stress in x-y plane
xxx
σhyd - hydrostatic pressure
νmax - maximum equivalent tangent Poisson�s ratio for concrete
µmax - maximum dilation rate
µu - ultimate dilation rate
aij - coeficients of deformation for stress concentration analysis
Eθ - modulus tangent to cutout
p - far field stress for stress concentration analysis
fcr - cracking stress for concrete
εco - strain at maximum stress for unconfined concrete
εMx - mechanical strain in x direction
εMy - mechanical strain in y direction
γMxy - mechanical shear strain in x-y plane
εTr - radial strain due to temperature change
εTH - hoop strain due to temperature change
γoxy - mid-plane shear strain
Vectors
{εo} - mid-plane strains
{κ} - curvatures
{σ1} - stresses in material coordinate system
{ε1}- strains in material coordinate system
{σx} - stresses in structure coordinate system
xxxi
{εx} - strains in structure coordinate system
{N} - section forces
{M} - section moments
Matrices
[A] - in-plane stiffness matrix
[A*] - in-plane flexibility matrix = [A]-1
[B] - in-plane out-of-plane coupling matrix
[D] - out-of-plane stiffness matrix
[Q] - ply stiffness matrix in material coordinate system
[ ]Q - ply stiffness matrix in structure coordinate system
xxxii
ABSTRACT OF THE DISSERTATION
Structural Characterization of Concrete Filled
Fiber Reinforced Shells
by
Andrew Davol
Doctor of Philosophy in Engineering Sciences (Structural Engineering)
University of California, San Diego, 1998
Professor Frieder Seible, Chair
Optimizing structures often leads engineers to combine several materials into a
hybrid system which utilizes the advantages inherent in each of the constituents.
Reinforced concrete is a classic example of such a system combining the superior
tension carrying capability of steel with the compression capacity and low cost of
concrete. A similar concept is being investigated which replaces the steel in a
conventional reinforced concrete member with a fiber reinforced polymer (FRP) shell.
These shells are manufactured with continuous relatively stiff fibers imbedded in a
softer matrix material. The nature of these materials allows the properties in various
xxxiii
directions to be controlled by placing fibers with prescribed orientations permitting the
engineer to �tailor� the material to a specific application. It is felt that such a system
may lead to efficient construction techniques that could reduce erection times and
construction costs due to the light weight of the FRP shells. This document examines
the structural behavior of concrete filled FRP shells concentrating on compression and
bending behavior, thermal response and stress concentration effects. Analytical
models are proposed to predict the stress and deformation state of the fiber reinforced
shell and concrete core under various loading conditions. The nonlinear response of
concrete confined by a linear elastic shell under compressive loads is investigated.
Experimental validation and calibration of these models has been carried out and is
presented. Related documents associated with this project explore the joining of
advanced composite components and the behavior of structural systems assembled
with these components.
1
1. INTRODUCTION
Structural engineers have long known the value of combining materials into a
composite structural system that takes advantage of the strengths inherent in each of
its constituents. Steel reinforced concrete is a classic example of this type of structural
system. These materials complement each other well due to the compression carrying
capability of the concrete and the tension carrying capability of the steel
reinforcement. Through the years this system has been improved upon as the
understanding of the concrete, steel and the interaction between the two has increased.
These improvements include the realization of the importance of transverse steel
reinforcement to provide confinement for the concrete core and to increase the shear
carrying capability of the structural member. It has also been well established that the
load carrying behavior of concrete in one principal direction is greatly affected by the
presence of stresses (or deformations) in the other principal directions. Specifically it
has been shown that the strength and ductility capacity can be greatly increased by the
presence of triaxial compression [1]. Such a triaxial state of compression is achieved
by providing �confinement� for the concrete. Under uniaxial compression concrete
will expand normal to the loading direction due to the Poisson�s effect and
microcracking. If reinforcement is placed to resist (confine) this expansion, the desired
stress state of triaxial compression is achieved. In traditional reinforced concrete
structures this confinement is attained by providing transverse reinforcement that takes
the form of hoops, spirals or stirrups as shown in Figure 1-1. This enhanced strength
2
and ductility become very important when considering a structure's ability to
withstand seismic loading. In many older structures insufficient transverse
reinforcement was provided to achieve the strength and ductility required to resist
seismic deformation demands. Brittle shear failures and ductile failures due to
insufficient confinement of the concrete core have been documented in inadequately
confined concrete members [2]. Retrofit measures have been developed to remedy
these shortcomings [2]. One common retrofit measure consists of placing a jacket
around a column to provide the lacking shear strength and or confinement. Steel
jackets have been successfully implemented for this purpose [2]. The need to custom
manufacture a steel shell for each column to be retrofitted and the time necessary to
weld the jackets in place has led to the development of alternate advanced composite
wraps such as fiber reinforced polymers (FRP) that are applied to the columns and
avoid the need for custom manufactured jackets and can speed installation procedures
[2]. These retrofit measures have led to the development of a new construction
concept or system that replaces the steel in a conventionally reinforced concrete
member with a premanufactured fiber reinforced shell as shown in Figure 1-. For this
new system the shell takes over the tension carrying, shear and confining actions
previously provided by the steel reinforcement and the concrete and shell combine for
compression load transfer as shown in Figure 1-2. The concept being investigated in
this project proposes using modular premanufactured fiber reinforced composite
shells, set in place on site and then filled with concrete. This system offers the
potential for substantial weight reduction as well as significantly reduced erection
3
times from those for current reinforced concrete structures as no removable forms or
heavy lifting equipment are required.
The premanufactured FRP shells are composed of relatively stiff fibers
embedded into a softer matrix material. The fiber orientations in the composite are
controlled to give the desired strength and stiffness in specified directions. The
materials and technologies associated with these composites are not new to structural
engineering. Aerospace structural engineers have long been taking advantage of the
tailorable qualities of these light weight materials. Cost concerns and lack of design
information consistent with civil engineering design practice have kept civil structural
engineers from serious consideration of these advanced composites but new materials
and manufacturing methods are under development that may change the cost equation
enough for these composites to become practical alternatives to conventional
structural systems. Furthermore, the constituent materials of the shell if chosen
properly and used in a manufacturing process with quality control can offer good
environmental resistance and longevity.
The shells are filled with concrete that is used to carry compression loads and
to stabilize the shell against buckling in compression as well as to aid in the joining of
adjacent members. This system has the ability to take great advantage of the concrete
core due to the confinement provided to the entire concrete core by the shell (no cover
concrete) and the linear elastic nature of the shell which can control the dilation of the
core much more than a ductile confining material such as steel.
4
This work represents one part of a program that was designed to prove the
feasibility of this new concept and to establish some initial design guidelines for its
implementation. The current document presents analytical models to characterize the
structural behavior of concrete filled FRP shells. This characterization is mainly
concerned with predicting the full stress state in the shell for all loading combinations
including thermally induced stresses, establishing rational failure criteria on which
design allowables can be based, investigating various material combinations and
investigating stress concentrations. Experimental investigations along these lines are
presented to verify the analytical modeling and to prove the viability of this concept.
Other related documents from this program address the implementation of the
advanced composite shell concept for new bridge structures [3] and joining concepts
[4] [5]. A prototype structure under experimental evaluation is pictured in Figure 1-3.
A review of the state of the art in areas pertinent to this analysis is presented in
the following chapter.
5
Figure 1-1 Conventionally Reinforced Concrete Column
Figure 1-2 Carbon Fiber Shell for Full Scale Testing
6
Figure 1-2 Concept for Concrete Filled Fiber Reinforced Shell
Figure 1-3 Prototype Bridge Structure With Carbon Shell Girders and a Fiber
Glass Deck System
7
2. RESEARCH REVIEW
The use of advanced composite wraps for retrofit measures has led to
considerable study in the area of concrete confined with linear elastic composite
wraps. At the University of California, San Diego, design guidelines have been
developed for advanced composite column retrofits [2]. The design guidelines are
based on supplying the column with the lacking transverse reinforcement necessary to
withstand seismic attack. These design equations were extended to a concrete filled
filament wound carbon shell system by Seible, Burgueño, Abdallah and Nuismer [6].
The design models make no effort at predicting the actual radial strain in the shell
throughout the loading. Compression tests on concrete filled advanced composite
shells have shown that large strength and ductility enhancements are possible with
these systems. Hoppel, Bogetti, Gillespie, Howie and Karbhari [7] investigated these
effects and proposed a Hooke�s law relation between the hoop strain in the shell, the
confining pressure and the axial stress in the concrete. Mirmiran and Shahaway [8]
proposed an incremental approach utilizing a cubic relation describing the change in
radial strain as a function of the axial strain. The coefficients of this cubic relation
were determined based on the unstressed and ultimate state (failure of the shell). This
model incorporated a variable Poisson�s ratio for the enclosed concrete based on a
model proposed by Elwi and Murray [9] which has been used for finite element
modeling of concrete [10]. The variable Poisson�s ratio was derived from compression
tests on unconfined concrete cylinders. The confining pressure in this approach is
8
calculated from the jacket hoop modulus, thickness, diameter and the radial expansion
of the core. Once the confining pressure is known, a constant pressure confinement
model is used to predict the concrete axial stress. This type of model works well for
shells with predominantly hoop fibers, since this architecture leads to shells with low
axial stiffness and a low Poisson�s ratio for loading in the axial direction. A similar
concept has been presented by Picher, Rochete and Lassiere [11] in the form of an
�effective confinement stiffness� based on the jacket hoop modulus, thickness and
diameter. These models do not work well for composite architectures including fibers
oriented away from the hoop direction. This was demonstrated in the extreme case by
tests performed by Howie and Karbhari [12]. In these tests various architectures were
investigated including several with all ±45o fibers. This lay-up leads to a very high
Poisson�s ratio for loading in the axial direction which was evident in the test results
as no increase in the strength or ductility of these specimens was observed when
compared to the unconfined concrete control specimens. The shell in this
configuration simply expands faster than the confined concrete core and offers no
resistance to the cracking in the concrete.
Bending of members using a composite shell for transverse reinforcement and
steel longitudinal reinforcement as in a conventional column was studied by Mirmiran,
Kargahi, Samaan and Shahaway [13].
To be able to quantify the physical state of the composite shell under all
loading conditions it is necessary to understand the deformation of the concrete core
when put under load. In the case of an unconfined concrete cylinder it has been shown
9
that the volumetric strain (εv), which is an invariant quantity defined as the sum of
three orthogonal normal strains (see Figure 2-1), reaches an absolute minimum
(maximum volume contraction) and then reverses until the net volume strain goes
through zero at an axial strain of approximately 80-100% of the strain reached at
maximum stress for an unconfined cylinder (εco). Beyond this level volume expansion
seems to increase unrestrained (Pantazopoulou [14]). Work by Mirmiran and
Shahaway [15] has shown that steel transverse reinforcement does delay this volume
expansion but does not prevent it because at higher radial strain levels the steel
reinforcement has yielded and no increase in confining pressure occurs with increasing
dilation. In the case of a linear elastic shell the increase in pressure with expansion
throughout the loading can prevent the volume expansion from occurring as seen in
Figure 2-2. Based on compression tests done at the University of Central Florida
Mirmiran shows that the dilation rate µ, defined as the incremental change in radial
strain divided by the incremental change in longitudinal strain (tangent Poisson�s
ratio), increases from the initiation of loading to a maximum value µmax at an axial
strain close to the ultimate axial strain of the unconfined concrete and then decreases
until it stabilizes at an ultimate value µu. The maximum and ultimate values are
correlated empirically to the jacket hoop stiffness and the concrete strength. In these
tests the composite shell thickness as well as the concrete strength was varied but the
shell architecture was held constant with a wrap angle of ±75o from the longitudinal
direction.
10
To predict the behavior of noncircular sections under compression loading it
becomes necessary to use approximate methods such as finite element analysis to
account for the variation in the confinement of the concrete core across the section.
Finite element modeling of concrete continues to be a much studied topic as many
difficulties arise from the nonhomogeneous anisotropic behavior once cracking
begins. Many models have been proposed through the years [16]. One of the earlier
models used was based on the Drucker-Prager soil mechanics yield surface which
expands the yield capacity of the material based on the current hydrostatic pressure
[17]. Some success has been demonstrated with this model for square cylinders with
rounded corners confined by linear elastic shells [18].
At the time of this writing no published material has been found on the
bending behavior of concrete filled FRP shells, however much work has been
published on concrete filled steel tubes (CFT). The use of the CFT system for
building and bridge columns has been extensively investigated since the 1950�s. Many
of the design approaches put forth are conservative and ignore the contribution from
the concrete fill. Furlong proposed a model in 1968 [19] which considered the
concrete and steel separately and added the components to get the system behavior for
axial and flexural loading. Tomii, Sakino, Watanabe and Xiao [20][21] investigated
short columns utilizing steel shells locally for added shear reinforcement and
confinement. Tomii and Sakino proposed a modification to a standard hoop reinforced
concrete confinement model proposed by Park for the encased concrete. The steel
shell was transformed into an equal number of spiral hoops. The modification was
11
used to slightly lower the predicted ultimate stress in the concrete to match
experimental data. Lu and Kennedy [22] performed compression tests on CFTs and
found that the addition of the contributions from the steel and unconfined concrete
individually gives a good estimate of the behavior of the composite section. This
makes sense as the shell will initially expand faster than the concrete core due to its
higher Poisson�s ratio leading to very little confinement of the concrete core.
This document extends the previous works by including the effects of the
Poisson�s ratio of the shell, establishing relations for the nonlinear concrete response
in compression and predicting the complete biaxial stress state in the shell under
flexure including the shear strains.
The constituent materials used in this system along with pertinent relations for
their analysis are described in Chapter 3.
12
P
1
2 3
Undeformed Cylinder
Deformed Cylinder
Area Strainε ε εA = +2 3
Volume Strainε ε ε εv = + +1 2 3
Figure 2-1 Area and Volume Strain Definition
Longitudinal Strain, ε1
-0.010-0.008-0.006-0.004-0.002
Vo
lum
etri
c S
trai
n, ε
v
-0.005
-0.004
-0.003
-0.002
-0.001
0.000
0.001
0.002 Elastic (1-2ν)ε1
Unconfined ConcreteLinear Elastic ConfinementMild Steel Confinement
Figure 2-2 Expansion Behavior of Concrete
13
3. MATERIAL CHARACTERIZATION 3.1 Advanced Composite Shells
Advanced composite materials have been used extensively in the aerospace
and defense sectors for over thirty-five years. The materials discussed in this
document consist of stiff fibers embedded in a relatively soft matrix material. These
materials are generally broken down into short fiber composites and continuous fiber
composites. Only the latter will be considered as the former usually do not have
sufficient stiffness for primary structural applications. This study is limited to
composites that consist of structures assembled from individual plies or lamina. An
individual lamina is composed of unidirectional fibers in a continuous matrix. These
laminae are stacked with varying fiber orientations to obtain the desired properties of
the assembled laminate. A brief description of the fibers and matrices used in this
study is presented in the following sections.
3.1.1 Fiber Reinforcement
There are several fiber reinforcement materials being investigated for use in
civil applications. The most promising at this time are E-glass, popular due to its low
cost and availability, and carbon useful for its excellent stiffness. Other popular fiber
reinforcements include aramids (Kevlar) which have high strength and stiffness but
experience problems with moisture absorption and polyester fibers that do not
generally possess sufficiently high stiffness for structural applications. Although other
fibers are available only carbon and E-glass will be considered here.
14
Glass fibers have been in use for engineering applications since the early
1940s. Several different types of glass fiber are commercially available. The most
common of these is designated E type. E-glass has low alkali content that attempts to
ensure corrosion resistance and high electrical resistivity. The major drawback to E-
glass for use in civil applications is that it has been reported to show poor chemical
resistance in both acidic and alkaline solutions which makes it a poor choice where it
is in contact with cement. A stiffer and stronger variant of E-glass was developed and
given the designation S-glass. It however still shares the problems of E-glass. An
alkali resistant fiber was developed and designated AR-glass or Z-glass but has had
limited success in practice. Table 3-1 gives some pertinent mechanical properties of
these fibers.
Table 3-1 Typical Properties of Commercial Glass Fiber Reinforcements [23]
Type of Fiber
Specific Gravity
Coef. Therm. Exp.
x10-6 oC-1
Young�s Modulus
GPa (msi)
Tensile Strengtha
GPa (ksi)
Poisson�s Ratio
E 2.54 5 72.4-76 (10.5-11)
3.6 (520)
.21
AR (Z) 2.68 7.5 70-80 (10.2-11.6)
3.6 (520)
.22
S 2.48 2.9-5.0 86 (12.5)
4.6 (667)
b
aVirgin Strength values. Actual strength values prior to incorporation into composite are ~2.1 GPa (305 ksi). bValue varies widely with changes in the manufacturing process.
Carbon fibers are attractive for civil structural applications due to their high
stiffness and low density along with their resistance to chemical attack. A wide range
of mechanical properties are available based on the precursor used to manufacture the
15
fibers and the manufacturing process itself. Table 3-2 shows some typical properties
for commercially available carbon fibers, Type I produced for high stiffness and Type
II for high strength. Note that the coefficient of thermal expansion is different in the
longitudinal and transverse directions due to the nonisotropic nature of the carbon
fibers.
Table 3-2 Mechanical Properties For Select Carbon Fibers [24]
Type of Fiber
Specific Gravity
Coef. Therm. Exp.
x10-6 oC-1 Longitudinal Transverse
Young�s Modulus
GPa (msi)
Tensile Strength
GPa (ksi)
TYPE I 1.95 -0.5 to -1.2 7-12
390 (56.6)
2.2 (319)
TYPE II 1.75 -0.1 to -0.5 7-12
250 (36.3)
2.7 (392)
3.1.2 Matrix Materials
The matrix material is used to bind the fibers and enable them to be combined
to create a composite. One of the most prominent composite materials currently in use
is the family known as polymeric composites or reinforced polymers. The matrix
materials used in these composites fall into two main classes, thermosetting and
thermoplastic resins. Thermoplastic resins are mainly used in short fiber applications
and preimpregnated composite plies and will not be discussed further here.
Thermosetting resins are predominant in continuous fiber applications and will be
briefly described below.
16
Common thermosetting resins used for composite material applications are
epoxy and polyester resins. The final product is produced by converting the liquid
resin into a solid through chemical cross-linking which leads to a tightly bound three-
dimensional network of polymer chains. Curing can be achieved at room or ambient
temperature but elevated temperature cure cycles can also be used. Thermosetting
resins are usually isotropic and do not melt on reheating. Table 3-3 presents some
typical properties for epoxy and polyester resins.
Table 3-3 Mechanical Properties For Common Thermosetting Resins [24] [25]
Type of Fiber
Specific Gravity
Coef. Therm. Exp.
x10-6 oC-1
Young�s Modulus
GPa (msi)
Tensile Strength
MPa (ksi)
Poisson�s Ratio
Epoxy 1.1-1.4 60 3-6 (.44-.87)
35-100 (5.1-14.5)
.38-.4
Polyester Vinylester
1.2-1.5 100-200 2-4.5 (.29-.65)
40-90 (5.80-13.1)
.37-.39
Epoxies are used extensively in structural applications due to the broad range
of mechanical properties that can be achieved. Elevated temperature curing cycles are
commonly used but are not required. Polyesters are the most widely used thermoset
resin system accounting for about 75% of the total resin used. They are used
extensively in marine applications and cure relatively quick at ambient temperature
through the addition of a catalyst. Vinylester are often considered to be part of the
polyester family. They were developed to combine the advantages of epoxies with the
faster cure of polyesters.
17
3.1.3 Manufacturing Processes
Many manufacturing methods have been developed to combine the fibers and
matrices into a finished composite part [26]. In the aerospace industry preimpregnated
plies (fibers preimpregnated with the uncured resin and then stored at low
temperatures to retard curing) are combined and then put under elevated temperatures
and pressures to achieve high fiber volume fractions (percentage of total volume of the
composite occupied by the fibers). This method is expensive and does not hold great
promise for civil applications. Methods more suited to civil applications combine the
resin and fibers as the part is being manufactured. Those commonly used with
continuous fibers include: (1) hand-lay-up where the dry fibers are placed at the
desired orientation and the resin is applied by hand and squeegeed into the fibers to get
complete coverage or wetting of the fibers, (2) filament winding where the dry fibers
are taken through a resin bath and wound onto a mandrel with the desired orientation
and (3) pultrusion which is similar to an extrusion process for metals with the
exception of having the part pulled through a die rather than pushed. These processes
all have limitations on the fiber volume fraction that can be achieved as well as the
fiber orientations possible. The shells used for the tests in this program were
manufactured with the filament winding process which limited the allowable angle
from the longitudinal axis for the fibers to greater than 10o [27].
3.1.4 Typical Ply Properties
Table 3-4 lists typical properties of unidirectional-fiber-reinforced epoxy
resins. These properties are strongly influenced by the fiber volume fraction. The E-
18
Glass properties listed in Table 3-4 represent a composite with a fairly low fiber
volume fraction most representative of a hand lay-up process whereas the carbon
properties are more representative of an aerospace quality preimpregnated ply.
3.1.5 Classical Lamination Theory
Classical lamination theory is the name given to the analytical methods used to
predict the behavior of a laminated composite material. The analysis assumes a
continuous displacement field through the thickness of the laminate which implies a
perfect bond between adjacent plies (no slip). A thin plate assumption is also used
which ignores the shear deformation of the composite. Thus a line initially
perpendicular to the mid-plane of the plate remains straight and perpendicular to the
mid-plane after deformation. It is also assumed that the through-the-thickness strains
are negligible. A description of the pertinent relations used in this work are included
here. Any text on mechanics of composite materials [28] [29] will give a full
derivation of the following relations.
The plate is assumed to lie in the x-y plane. If a point on the mid-plane of the
undeformed section is displaced by uo, vo and wo in the x, y and z directions
respectively the deformation of any point is given by,
u u z wx
v v z wy
w w
oo
oo
o
= −
= −
=
∂∂
∂∂
. (3-1)
19
Table 3-4 Typical Ply Properties for Fiber-Reinforced Epoxy Resins [28]
Property E-Glass Carbon Fiber Volume Fraction 46 63 Specific Gravity 1.8 1.61 Tensile Strength, 0o MPa
(ksi)1104 (160)
1725 (250)
Tensile Modulus, 0o GPa(msi)
39 (5.66)
159 (23.1)
Tensile Strength, 90o MPa(ksi)
36 (5.22)
42 (6.09)
Tensile Modulus, 90o GPa(msi)
10 (1.45)
10.9 (1.58)
Compression Strength, 0o MPa(ksi)
600 (87.0)
1366 (198)
Compression Modulus, 0o GPa(msi)
32 (4.64)
138 (20.0)
Compression Strength, 90o MPa(ksi)
138 (20.0)
230 (33.4)
Compression Modulus, 90o GPa(msi)
8 (1.16)
11 (1.60)
In-Plane Shear Strength MPa(ksi)
95 (13.8)
In-Plane Shear Modulus GPa(msi)
6.4 (0.93)
Longitudinal Poisson�s Ratio (νLT) 0.25 0.38 Longitudinal Coef. of Thermal Expansion (10-6/oC)
5.4 0.045
Transverse Coef. of Thermal Expansion (10-6/oC)
20.2 36
20
The strains can be derived from the assumed displacement field as
ε ∂∂
∂∂
ε ∂∂
∂∂
γ ∂∂
∂∂
∂∂ ∂
xo o
yo o
xyo o o
ux
z wx
vy
z wy
uy
vx
z wx y
= −
= −
= + −
2
2
2
2
2
2
. (3-2)
Defining mid-plane strains as
{ }εεεγ
∂∂∂∂
∂∂
∂∂
oxo
yo
xyo
o
o
o o
uxvy
uy
vx
=�
��
��
�
��
��
=
+
�
�
���
�
���
�
�
���
�
���
, (3-3)
and mid-plane curvatures as
{ }κκκκ
∂∂
∂∂∂∂ ∂
=�
��
��
�
��
��
= −
�
�
���
�
���
�
�
���
�
���
x
y
xy
o
o
o
wxwy
wx y
2
2
2
2
2
2
, (3-4)
we can then write the general strains as
εεγ
εεγ
κκκ
x
y
xy
xo
yo
xyo
x
y
xy
z�
��
��
�
��
��
=�
��
��
�
��
��
+�
��
��
�
��
��
. (3-5)
Each individual lamina being composed of unidirectional fibers embedded in a
matrix can be modeled as a transversely isotropic material, with the plane of isotropy
21
being normal to the fiber direction. If we define a local coordinate system (1,2,3) for
the lamina with the 1 direction parallel to the fiber direction and the 2 and 3 directions
normal to the fiber direction we can then write the in-plane stress strain relations for a
given lamina as
{ } [ ]{ }σ ε1 1= Q , (3-6)
with the stress vector given by
{ }σ
σσστττ
1
1
2
3
23
13
12
=
�
�
���
�
���
�
�
���
�
���
(3-7)
and the strain vector given by
{ }ε
εεεγγγ
1
1
2
3
23
13
12
=
�
�
���
�
���
�
�
���
�
���
. (3-8)
The nonzero terms of the 3x3 stiffness matrix for the in-plane behavior are given by
22
( )
( )
( )
QE
QE
QE
Q G
111 23 32
12 21 23 32 13 31 12 23 31
222 13 31
12 21 23 32 13 31 12 23 31
122 12 13 32
12 21 23 32 13 31 12 23 31
66 12
11 2
11 2
1 2
=−
− − − −
=−
− − − −
=+
− − − −=
ν νν ν ν ν ν ν ν ν ν
ν νν ν ν ν ν ν ν ν ν
ν ν νν ν ν ν ν ν ν ν ν
, (3-9)
with E1 the modulus of the composite lamina in the fiber direction, E2 the modulus
normal to the fiber direction, νij the Poisson�s for loading in the i direction and G12 the
in-plane shear modulus. This material stiffness matrix is valid for a coordinate system
aligned with the material directions. For a coordinate system oriented arbitrarily to the
material coordinate system it becomes necessary to transform the stiffness matrix,
from the orthogonal material coordinate system (1,2,3) to the orthogonal structure
coordinate system (x,y,z). The transformation between the two is given by
[ ]123
�
��
��
�
��
��
=�
��
��
�
��
��
Txyz
. (3-10)
If the 3 and z directions are parallel and the x,y coordinate system is rotated relative to
the 1,2 coordinate system by an angle θ (see Figure 3-1), [T] is the 3x3 transformation
matrix shown in equation 3-11.
23
1
2
3,z
xy
Figure 3-1 Material and Structural Coordinate Systems
[ ]T = −− −
�
�
���
�
�
���
cos sin sin cossin cos sin cos
sin cos sin cos cos sin
2 2
2 2
2 2
22
θ θ θ θθ θ θ θ
θ θ θ θ θ θ (3-11)
The stresses and strains are then transformed as
[ ]σστ
σστ
1
2
12
�
��
��
�
��
��
=�
��
��
�
��
��
Tx
y
xy
(3-12)
and
[ ]εε
γ
εε
γ
1
2
1212
12
�
�
��
�
��
�
�
��
�
��
=
�
�
��
�
��
�
�
��
�
��
Tx
y
xy
. (3-13)
The stress strain relation in the structure coordinate frame (x,y,z) is then expressed as
{ } [ ]{ }σ εx xQ= , (3-14)
where
θ
24
[ ] [ ] [ ][ ]Q T Q T= −1 (3-15)
and
{ }
{ }
σσστ
εεεγ
x
x
y
xy
x
x
y
xy
=�
��
��
�
��
��
=�
��
��
�
��
��
. (3-16)
The nonzero terms of the stiffness matrix in the structure coordinate frame are given
as
[ ]
[ ]
Q Q Q Q QQ Q Q Q Q
Q Q Q Q Q
Q Q Q Q Q
Q Q Q Q Q
Q Q
11 114
224
12 662 2
12 11 22 662 2
124 4
16 112
222
12 662 2
22 224
114
12 662 2
26 112
222
12 662 2
66 662 2 2
2 24
2
2 2
2
= + + +
= + − + +
= − − + −
= + + +
= − + + −
= − +
cos sin ( ) cos sin( ) cos sin (cos sin )
cos sin ( )(cos sin ) cos sin
cos sin ( ) cos sin
sin cos ( )(cos sin ) cos sin
(cos sin )
θ θ θ θθ θ θ θ
θ θ θ θ θ θ
θ θ θ θ
θ θ θ θ θ θ
θ θ ( ) cos sinQ Q Q11 22 122 22+ − θ θ
. (3-17)
The ply stress-strain relation shown in equation 3-14 can now be used to derive
the resultant forces and moments in the laminated plate by integrating through the
thickness of the laminate. The geometry of the plate is defined in Figure 3-2.
25
12
k
n
t
x
z
h0h1 h2
hkhn-1
hn
midplane
laminanumber
Figure 3-2 Geometry of Laminate
The section forces and moments can now be written as
{ }
{ }
NNNN
dz
MMMM
zdz
x
y
xy
x
y
xyt
t
x
y
xy
x
y
xyt
t
=�
��
��
�
��
��
=�
��
��
�
��
��
=�
��
��
�
��
��
=�
��
��
�
��
��
−
−
�
�
σσσ
σσσ
/
/
/
/
2
2
2
2
. (3-18)
These integrals can be handled as summations of the individual layers of the laminated
plate.
{ } [ ] { } [ ] { }
{ } [ ] { } [ ] { }
N Q dz Q k zdz
M Q zdz Q k z dz
ko
kh
h
h
h
k
n
ko
kh
h
h
h
k
n
k
k
k
k
k
k
k
k
= +���
��
���
��
= +���
��
���
��
−−
−−
���
���
=
=
ε
ε
11
11
1
2
1
(3-19)
Combining the stress and section force definitions the relation between the section
forces and deformations is derived.
26
NNNMMM
A BB D
x
y
xy
x
y
xy
xx
yy
xy
x
y
xy
�
�
����
�
����
�
�
����
�
����
=�
�
�
�
�
����
�
����
�
�
����
�
����
εεγκκκ
, (3-20)
with
( ) ( )
( ) ( )
( ) ( )
A Q h h
B Q h h
D Q h h
ij ij k k kk
n
ij ij k k kk
n
ij ij k k kk
n
= −
= −
= −
−=
−=
−=
�
�
�
11
21
2
1
31
3
1
1213
, (3-21)
where [A], [B] and [D] are 3x3 matrices. The A matrix represents the in-plane force
strain relations. The B matrix represents the coupling between in-plane and out-of-
plane behavior (extension - bending). The D matrix represents the out-of-plane
behavior.
The analysis presented for circular shells only considers the in-plane properties
of the shell. This means that the coupling and bending matrices, curvatures and
moments do not enter into the calculations. As soon as the section considered is no
longer circular the bending behavior must be considered. If only the in-plane behavior
is of interest equivalent orthotropic plate properties can be determined as described in
the following section.
27
3.1.6 Equivalent Plate Properties
Assuming that only the in-plane behavior of the laminate is of concern
equivalent orthotropic plate properties can be derived for a symmetric laminate. These
equivalent properties will be used later in the analytical relations derived for circular
sections. For a symmetric laminate there is no shear extension coupling which gives
A16=A26=0. The in-plane flexibility relation is given as
εεγ
x
y
xy
x
y
xy
A AA A
A
NNN
�
��
��
�
��
��
=�
�
�
�
��
��
�
��
��
11 12
21 22
66
00
0 0
* *
* *
*
, (3-22)
with
( )( )
( )
A A A A A
A A A A A
A A A A A
A A
11 22 11 22 122
22 11 11 22 122
12 12 11 22 122
66 661
*
*
*
*
/
/
/
/
= −
= −
= − −
=
. (3-23)
The same relation for an orthotropic plate assuming plane-stress can be written as
εεγ
ν
νx
y
xy
x
xy
x
xy
x y
xy
x
y
xy
tE tE
tE tE
tG
NNN
�
��
��
�
��
��
=
−
−
�
�
�
�
��
��
�
��
��
1 0
1 0
0 0 1
. (3-24)
Equating equations 3-22 and 3-24 we can derive the equivalent orthotropic plate
properties as
28
E tA A A A tAE tA A A A tA
A tE A A
E E A A
G tA A t
x
y
xy x
yx xy y x
xy
= = −
= = −
= − =
= =
= =
11
1
11 11 22 122
22
22 11 22 122
11
12 12 22
12 11
66 66
/ ( ) ( ) // ( ) ( ) /
/
( / ) /
/ ( ) /
*
*
*
*
ν
ν ν
. (3-25)
3.1.7 First Ply Failure Criteria
For the analyses discussed in this document the shell will be defined to have
failed when one of its plies has exceeded its allowable stress in the fiber direction. To
determine the ply stresses it is necessary to transform the strains in the global
coordinate system into the local ply coordinate system from which the ply stresses can
be determined. For the analysis presented in this document the strains in the structure
coordinate system are known. Equation 3-14 is used to transform these strains to
stresses in the structural frame. Equation 3-12 is then applied to transform the stresses
into the material coordinate system. Once the stresses in the material coordinate frame
are known they can be compared to the ply allowables.
3.1.8 Thermal Expansion
Concerns have been raised based on the potential difference in the coefficient
of thermal expansion for the composite shell and the concrete core. As can be seen in
Table 3-1 and Table 3-2 the coefficient of thermal expansion for reinforcing fibers in
the fiber longitudinal direction can be very low or even negative. This low thermal
expansion coefficient for the fibers can lead to very small thermal strains being
29
induced in the laminated shell by a temperature change depending on the composite
architecture chosen.
At the lamina level these materials behave orthotropically under temperature
changes due to the orientation of the fibers in one direction and the difference in the
coefficients of thermal expansion for the fibers and the matrix. The thermal strains in
the longitudinal and transverse directions are given by the following equations:
ε αε α
1 1
2 2
T
T
TT
==
∆∆
, (3-26)
with
ε1T - the thermal strain in the fiber direction
ε 2T - the thermal strain in the transverse direction
α 1 - the coefficient of thermal expansion in the fiber direction
α 2 - the coefficient of thermal expansion in the transverse direction
∆T - temperature change
Equivalent thermal expansion coefficients for a given lamina in the structure
coordinate frame may be determined from the transformation
αα
α
θ θ θ θθ θ θ θ
θ θ θ θ θ θ
αα
x
y
xy12
22
0
2 2
2 2
2 2
1
2
�
�
��
�
��
�
�
��
�
��
=−
− −
�
�
�
�
��
��
�
��
��
cos sin sin cossin cos sin cos
sin cos sin cos cos sin, (3-27)
30
where αxy is an apparent coefficient of thermal shear. With these coefficients the
thermal strains in the ply can be given by
εεγ
ααα
xT
yT
xyT
x
y
xy
TTT
�
��
��
�
��
��
=�
��
��
�
��
��
∆∆∆
. (3-28)
If free to expand no stresses would develop in the ply, however in a laminated
composite the plies are not free to expand due to the restraint offered from the adjacent
plies. The stresses generated from this restraint can be related to the mechanical strains
which are given as
εεγ
εεγ
ααα
xM
yM
xyM
x
y
xy
x
y
xy
TTT
�
��
��
�
��
��
=�
��
��
�
��
��
−�
��
��
�
��
��
∆∆∆
. (3-29)
The first strains on the right side of equation 3-29 are defined in Section 3.1.3. The
stress strain relation for a ply can now be expressed in terms of the mid-plane strains,
curvatures and thermal strains.
[ ]σστ
ε κ αε κ α
γ κ α
xT
yT
xyT
xo
x x
yo
y y
xyo
xy xy
Qz Tz Tz T
�
��
��
�
��
��
=+ −+ −+ −
�
��
��
�
��
��
∆∆∆
(3-30)
The laminate will deform under thermal loading with no external forces being applied.
Using equations 3-18 through 3.20 and 3-30 the following �thermal forces� are
obtained. These are the forces that if applied would give the same deformations as the
change in temperature.
31
[ ] ( )NNN
Tk
Q h hxT
yT
xyT
n
k
x
y
xy k
k k
�
��
��
�
��
��
= �=
�
��
��
�
��
��
− −∆1 1
ααα
(3-31)
[ ] ( )MMM
Tk
Q h hxT
yT
xyT
n
k
x
y
xy k
k k
�
��
��
�
��
��
= �=
�
��
��
�
��
��
− −12 1
21
2∆ααα
. (3-32)
Using these fictitious thermal forces and moments the mid-plane strains and
curvatures for the laminate can be calculated from equation 3-20. By carrying out this
analysis with a unit temperature change, equivalent coefficients of thermal expansion
can be determined for the laminate. This will result in not only expansional
coefficients but also shear and curvature coefficients if the lay-up is not symmetric.
For a symmetric lay-up there is no coupling between the in-plane strains and the
curvatures which allows the in-plane thermal expansion coefficients to be calculated
from the [A] matrix alone.
3.2 Concrete
Concrete is one of the most commonly used and commonly analyzed structural
civil engineering materials in use today and yet its response to applied loads is still not
fully understood. This difficulty in characterizing the mechanical behavior of concrete
is due to its highly nonhomogeneous structure and its nonlinear behavior due to crack
formation.
32
3.2.1 Uniaxial Compression
Although concrete is composed of materials that individually can be
considered to display brittle elastic behavior its stress strain curve is nonlinear. This
can be explained by the formation of microcracks as the loading progresses. The
volumetric strain defined as εv=ε1+ε2+ε3 (see Figure 2-1) initially decreases as would
be predicted by the theory of elasticity but almost immediately the volume strain
begins to deviate from this prediction and the volume begins to decrease at a slower
rate. When the concrete gets close to its ultimate strain the volume strain has actually
become positive (net volume expansion).
3.2.2 Biaxial and Triaxial Stress States - Confinement Effects
The response of concrete to bi- and triaxial stress states has been the subject of
many studies over the years. Kupfer, Hillsdorf and Rusch [30], Kupfer and Gerstle
[31], Gerstle et al. [32] and Vecchio and Collins [33] have studied concrete under
biaxial loading to determine the effects of an orthogonal load on the strength of
concrete. All researchers report an increase in the strength of concrete due to an
orthogonal compression stress and a decrease due to an orthogonal tension stress.
Kupfer and Gerstle have proposed a correlation between the strength increase and the
orthogonal confining stress. Veccio and Collins have proposed a relation based on the
orthogonal strain. This allows for the gradual reduction of strength observed based on
the orthogonal tensile strains.
33
Concrete under triaxial compression has been studied since early this century.
Hydraulic chambers and mechanical restraints have both been used to achieve the
desired stress state. Many models have been proposed to account for the increase in
strength and ductility observed in confined concrete. The main distinguishing point
between these models is the nature of the confining pressure. Some models assume a
constant radial pressure throughout the loading history. This simulates the case of steel
reinforcing spirals and hoops well since the steel yields fairly early on in the loading
and then produces a constant confining pressure proportional to the yield strength, area
and spacing of the transverse reinforcement. Mander has proposed such a model for
steel reinforced concrete [34][35]. Figure 3-3 shows the response of a typical concrete
cylinder under compression for various levels of confining pressure as predicted by
Mander�s model. The second type of model considers the variation in the confining
stress due to the actual dilation of the concrete core at any given point in the loading.
This second type of model works well for concrete confined by a linear elastic shell.
One popular method for formulating a confinement model dependent on the current
dilation state is to use a constant pressure model incrementally [36]. For each
increment of load the dilation of the system is calculated, knowing the dilation the
confining pressure is calculated and a constant pressure model can be used to predict
the current stress state. This is shown graphically in Figure 3-3 by the incremental
model which has increasing pressure as the axial strain increases. This causes the
incremental model to cross the constant pressure curves at points where the pressure in
the core equals that used for the constant pressure model.
34
The difficulty in pursuing this approach is the ability to accurately predict the
dilation of the confined concrete throughout the loading. Almost immediately upon
loading the concrete will begin to form microcracks and the dilation behavior will
deviate from that which would be predicted by Poisson�s effect in linear elasticity (see
Figure 2-2). The approach that is taken for the analysis described in Chapter 4 is
similar to the passive modeling approach described above except that the current
dilation state is used directly to estimate the tangent modulus of the confined concrete.
This approach is detailed in Section 4.1.1.2
Strain
0.000 0.005 0.010 0.015 0.020 0.025 0.030
Str
ess/
f'c
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5Mander's ModelIncremental Mander's Model
Unconfined
fl/f’c
0.05
0.10
0.20
0.40
Figure 3-3 Stress Strain Models for Confined Concrete
35
3.2.3 Tension
In general the tension strength of the concrete is ignored in the analytical
models used to predict the strength of the concrete filled FRP members studied in this
document. This is done to give a conservative estimate of the performance of this
system. The tension carrying capacity of the concrete may effect the stiffness of the
system even after the concrete is significantly cracked due to tension stiffening effects.
Studies will be presented to demonstrate the effects of including the tension carrying
contribution of the concrete on the system behavior. For the purpose of these studies
the following relations taken from Collins and Mitchell [37] will be used to estimate
the tension carrying capacity of the concrete. The cracking strength of the concrete for
stiffness evaluation, fcr, will be given by
f f MPa
f f psi
cr c
cr c
=
=
0 63
7 5
.
.
'
'
λ
λ . (3-26)
The λ factor is used to account for concrete density with
λ=1.00 for normal weight concrete
λ=0.85 for sand-lightweight concrete
λ=0.75 for all-lightweight concrete
The analytical models presented in Chapter 4 are based on a section analysis
approach that establishes a secant stiffness for the structural member at the current
loading state. This secant modulus is then used to predict member deformations under
load. As has been demonstrated in conventional reinforced concrete structures the
36
actual stiffness of a structural member will be greater than that predicted by a section
analysis approach due to the effect of tension stiffening. Tension stiffening is a
phenomenon that occurs due to the fact that the concrete cracks discretely on the
tension side of the member leaving sections of the member in between cracks intact or
uncracked. These uncracked sections will tend to stiffen the structure as load transfer
will still occur from the reinforcement into the uncracked concrete. To account for this
effect an average tension stress for the cracked concrete, fc, is used. This smeared
average concrete stress is approximated by
f fc
cr
cf
=+β β
ε1 2
1 500, (3-27)
where β1− factor accounting for bond characteristics of reinforcement
β1=1.0 for deformed reinforcing bars, or ribbed shells
β1=0.7 for plain bars, wire, strands or smooth shells
β1=0 for unbonded reinforcement
β2=factor accounting for sustained or repeated loading
β2=1.0 for short-term monotonic loading
β2=0.7 for sustained and/or repeated loading
εcf=average strain in section. The effect of this phenomenon will be studied in Chapter 8.
37
The material behavior presented in this section will be used in Chapter 4 to
develop the analytical models that describe the behavior of these concrete filled fiber
reinforced shells under various loading conditions.
38
4. ANALYTICAL MODELING OF CONCRETE FILLED FRP SHELLS 4.1 Circular Shells
The analysis presented in this section assumes that the shell is thin compared to
the radius of the section and that the strain gradient in the shell is negligible. Thus for
a circular section the bending stiffness of the laminate has no effect on the analysis
and the equivalent plate properties presented in Section 3.1.5 can be used. In this
section the compression and tension behavior are first derived. These relations are then
used to develop the bending behavior. An incremental elastic approach is taken for
this analysis. The fiber reinforced shell is assumed to remain linear elastic throughout
the loading. The tangent modulus and an equivalent tangent Poisson�s ratio of the
concrete are used in an incremental relation to introduce the nonlinear concrete
behavior.
4.1.1 Compression
4.1.1.1 Linear Elastic Relations
4.1.1.1.1 Shell Properties
For analysis purposes equivalent orthotropic properties for the laminated plate
are used to describe the shell behavior. Once a suitable lay-up is defined the lamina or
ply properties are used to derive equivalent plate properties through the use of
classical lamination theory as described in Section 3.1. This analysis results in
39
Young�s moduli for the longitudinal and hoop directions along with the associated
Poisson�s ratios. A biaxial stress state is assumed to exist in the shell. Thus the
longitudinal and hoop strains can be given respectively as
ε σ ν σL
L
LHL
H
HE E= − (4-1)
and
ε σ ν σH
H
HLH
L
LE E= − . (4-2)
4.1.1.1.2 Concrete Properties
The concrete model employed assumes that the compression response is
dependent on the radial expansion of the concrete. The longitudinal and radial strains
in the concrete are given respectively as
ε σ ν σ1 11 2= −
Ecc r( ) (4-3)
and
ε ν σ ν σrc
c r cE= − −1 1 1(( ) ) . (4-4)
4.1.1.1.3 System Behavior
The system is assumed to work together with no slip between the shell and the
concrete core. This is referred to as full composite action. Assuming full composite
action between concrete core and composite shell the following must hold.
40
ε εε ε
σ σ
L
H r
H rRt
==
= −
1
(4-5)
Utilizing equations 4-1 through 4-5 the radial pressure, radial strain and axial stress
can be determined.
σ ν ε εν νr
LH r H
HL LH
E tR
= − +−
( )( )
1
1 (4-6)
ε ε ν ν ν ν ν νν ν ν νr
c c HL LH LH H c c
H c c c HL LH
E R E tE t E R
= − + − −− − − − −
�
��
�
��1
2
2
1 1 21 2 1
( ) ( )( ) ( )
(4-7)
σ ε σ νL L L r
HL
H
EE
Rt
= −�
��
�
�� (4-8)
4.1.1.2 Nonlinear Concrete Response
The experimental data gathered from the confined cylinder tests described in
Section 5.1.2 was used to derive relations for the nonlinear concrete response. The
longitudinal and hoop strains are extracted directly from the strain gage and linear
potentiometer readings. The dilation rate µ is calculated from the strain data as shown
in equation 4-9.
µ εε
= ∆∆
r
1
(4-9)
The experimental dilation rate shows considerable scatter due to the fact that the
relation is defined incrementally. A curve fit of the form suggested by Mirmiran [8] as
41
shown in equation 4-10 is used to smooth the data (see Figure 4-1). In this relation
µ0 represents the initial dilation rate that is calculated from equation 4-7 with the
initial concrete Poisson�s ratio and modulus.
µµ ε
εεε
εε
εε
=+ +
�
��
�
��
+ +�
��
�
��
01 1
2
1 1
2
1
a b
c d
co co
co co
(4-10)
A second equation is derived from the test data relating the incremental change in the
concrete stress to the incremental axial strain as shown in equation 4-11.
σ
εε
εε
εε
1
1
1 12
1
=
+ +�
��
�
��
fa
b cc
co
co co
' (4-11)
The concrete stress is determined by subtracting the load in the shell from the total
applied load. Equations 4-10 and 4-11 are used in a program to simultaneously solve
equations 4-3 and 4-7 for the equivalent tangent Poisson�s ratio and tangent concrete
modulus as a function of the axial or radial strain. As discussed in Section 3.2.2 the
behavior of the confined concrete can be characterized by the confining pressure or by
the actual deformations. Models have been proposed that take the latter approach to
define a tangent or secant modulus for the concrete based on the current dilation or
�damage state� of the core [14]. For this analysis a curve fit of the form shown in
equation 4-12 was used to describe the tangent modulus as a function of the initial
modulus and the current radial strain.
42
E Ea
b c
c co
r
co
r
co
r
co
=
+
+ +�
��
�
��
1
12
εε
εε
εε
(4-12)
Longitudinal Strain
-0.04-0.03-0.02-0.010.00
Dila
tio
n R
ate
0.0
0.1
0.2
0.3
0.4
ExperimentalSmoothed
Figure 4-1 Experimental and Smoothed Dilation Rate
This evaluation was carried out for each of the compression cylinders tested. As
described in Chapter 5 two of the thick all hoop cylinders experienced considerable
bending and this data was considered compromised and was not utilized for the
analysis. The two helical cylinders had the hydrostone break away from the shell
during loading (see Chapter 5 for details) which made it impossible to accurately
determine the concrete stress at high radial strains. The analytical relation for the
concrete tangent modulus was based on the six remaining cylinders. The tangent
43
modulus vs. the radial strain for these six cylinders is shown in Figure 4-2 along with
the average tangent modulus. Table 4-1 gives the constants used for the analytical
model in equation 4-12 that represents the average of the test data.
Radial Strain
0.000 0.001 0.002 0.003 0.004
Co
ncr
ete
Tan
gen
t M
od
ulu
s (k
si)
0
500
1000
1500
2000
2500
3000
3500
Co
ncr
ete
Tan
gen
t M
od
ulu
s (M
Pa)
0
5000
10000
15000
20000
25000
AverageExperimental
Figure 4-2 Concrete Tangent Modulus vs. Radial Strain
Table 4-1 Constants for Tangent Modulus Relation
Constant a b c d Eco S.I. Eco U.S.
Value 1.0 -0.09967 5.802 7.061 20.9 (MPa) 3.03x106 (psi)
Figure 4-3 shows the equivalent tangent Poisson�s ratio vs. the radial strain for
the six cylinders analyzed. From this plot it can be seen that the maximum equivalent
tangent Poisson�s ratio for all cylinders occurs when the radial strain is approximately
44
0.2%. For the analytical model it has been assumed that the maximum equivalent
tangent Poisson�s ratio is a function of the ratio of the hydrostatic pressure (σhyd) in the
concrete when the radial strain reaches 0.2% to the unconfined concrete strength. A
cubic polynomial is used to approximate the relation between the equivalent tangent
Poisson�s ratio and the radial strain. The initial slope of these curves was observed to
be constant (~200) for all cylinders tested and will be represented by νci�. The initial
value at no radial strain is simply the initial concrete Poisson�s ratio. At a radial strain
of 0.002 the value of the polynomial must be νmax and the slope of the curve must be
0. These conditions are sufficient to establish the constants of the polynomial as
shown in equation 4-13. To simulate the observed behavior that the dilation rate seems
to flatten out the Poisson�s ratio is assumed to stay constant after a radial strain of
0.0025 has been reached. This value was based on the compression test data. Figure
4-4 compares the analytical model to the experimental curves.
ν ν ε ε εν
ν ν νν
c co r r r
ci
co ci
ci
b c dbcd c e
= + + +
=
= − −
= − −
2 3
750 000 100033333 8 333 4
'
max'
'
, ( ). .
(4-13)
Figure 4-5 shows the variation of the maximum Poisson�s ratio with
hydrostatic pressure. At this time this relation is based on scant data especially in the
region of low confinement. The maximum value seen in Figure 4-5 of 0.5 for the
maximum tangent Poisson�s ratio is used here for lack of any reliable data in this
range. From Figure 2-2 it can be seen that a value higher than 0.5 must be possible for
45
the slope of the volume strain curve to be positive as is the case for unconfined
concrete. More experimental investigation in this range is needed to accurately predict
the behavior for low levels of confinement. A relation of the form shown in equation
4-14 defines the analytical curve used for this analysis. This relation is the last piece
needed to formulate the compression behavior.
νσ σ
max ' '. . .= + −�
��
�
��0 488 0 047 0 035
2hyd
c
hyd
cf f (4-14)
The flow for the analytical model to predict the compression behavior is
depicted schematically in Figure 4-6. The determination of the maximum equivalent
tangent Poisson�s ratio is shown in Figure 4-7.
The above findings demonstrate the connection between the dilation and load
carrying capacity of the concrete core. Since the tangent modulus is based on the
current dilation state of the core, delaying this dilation will enhance the load carrying
capacity of the core. The critical variables for offering maximum dilation control are
the stiffness of the shell in the hoop direction and Poisson�s ratio of the shell for
loading in the longitudinal direction. A steel shell while offering high stiffness in its
linear range also has a high Poisson�s ratio meaning that in the initial stages of loading
the shell will expand faster than the concrete core. When the steel shell reaches its
yield strain, which for mild steel occurs at a radial strain of approximately 0.2%, the
hoop stiffness essentially becomes zero allowing the expansion of the core to increase
and eventually the net volume strain will become positive as described in Chapter 2.
46
Radial Strain
0.000 0.001 0.002 0.003 0.004 0.005
Eq
uiv
alen
t T
ang
ent
Po
isso
n's
Rat
io
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
Thin Shells ExperimentalThick Shells Experimental
Figure 4-3 Equivalent Tangent Poisson's Ratio for Test Cylinders
Radial Strain
0.000 0.001 0.002 0.003
Eq
uiv
alen
t T
ang
ent
Po
isso
n's
Rat
io
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
Thin Shells ExperimentalThick Shells ExperimentalThin Shells AnalyticalThick Shells Analytical
Figure 4-4 Analytical Equivalent Tangent Poisson's Ratio
47
σhyd/f'c1.0 1.5 2.0 2.5
Max
imu
m E
qu
ival
ent
Tan
gen
t P
ois
son
's R
atio
0.40
0.42
0.44
0.46
0.48
0.50
ExperimentalAnalytical
Figure 4-5 Maximum Tangent Poisson's Ratio vs. Hydrostatic Pressure
48
Increment axial strain
Calculate dεr from eqn (4-7)
εr=εr+dεr
Calculate tangent modulus Ec from eqn (4-2)
Calculate Poisson�s ratio from eqn (4-13)
Calculate orthotropic plate properties for shell
Calculate σL from eqn (4-8)
σ1=σ1+dε1 Ec Figure 4-6 Analytical Model For Compression Behavior
49
Increment axial strain
Guess max Poisson�s ratio vmax
Calculate dεεεεr from eqn (4-7)
εεεεr=εεεεr+dεεεεr
Define cubic relation for vc from eqn (4-13)
Calculate tangent modulus from eqn (4-12)
Calculate Poisson�s ratio from eqn (4-13)
Does εεεεr=.002?
Yes
Calculate σσσσhyd
Calc vmax_anal from eqn (4-14)
Does vmax =vmax_anal
No
Yes No
vmax =vmax_analContinue
Calculate orthotropic plate properties for shell
Figure 4-7 Determination of Maximum Equivalent Tangent Poisson's Ratio
50
4.1.2 Tension
In tension the longitudinal stress in the concrete core is taken to be zero. The
shell will tend to contract radially due to its Poisson�s ratio. This radial contraction is
resisted by the concrete core. The radial stress in the concrete is assumed to be
σ εr H coE= . (4-15)
Using this relation, equations 4-1, 4-2 and the last of equations 4-5 we can derive the
hoop strain as a function of the axial strain as
ε ε νν ν νH L
H LH
LH HL co H co LH
E tE R E t E R
=− −
�
��
�
�� . (4-16)
This relation is linear and mainly dependent on the Poisson�s ratio and the hoop
modulus of the shell as well as the initial Young�s modulus of the concrete. An
isotropic material such as steel would behave exactly the same as long as the material
remained linear.
4.1.3 Shear
Two cases will be discussed for determining the shear strains in the shell. The
first case applies to the condition of an inadequate or nonexistent shear transfer
mechanism between the shell and the encased concrete core. The second case applies
to a shell and concrete composite system that has an adequate shear transfer
mechanism. Figure 4-8 shows the forces in a section of shell and concrete necessary
to maintain equilibrium in a section of the member with shear forces present. The load
51
in the shell increases from P to P� over a defined distance. To maintain equilibrium the
shear in the shell at the section of interest, qs, plus the shear transferred between the
shell and the concrete core, qc, must equalize the difference between the shell loads. If
the shear mechanism provided between the shell and the concrete core is inadequate to
provide a shear transfer equal to qc then the core and shell will slip relative to each
other and the shear distribution between the shell and the concrete will be determined
by the relative shear stiffness of the members. If the shear transfer mechanism is
adequate to take the shear qc then the members will deform together and a plane-
sections-remain-plane assumption may be used.
Ps`
Ps
Pc
Pc` qs
qs
qc
qc
qc`
FRP Shell
Concrete Core
Figure 4-8 Shear Transfer Between Concrete Core and Composite Shell
52
For the first case the concrete and shell shear resistance must be considered
separately. The initial shear stiffness of the concrete core is much greater than the
initial shear stiffness of the shell. It is therefore assumed that the concrete takes the
entire shear until it has reached its shear carrying capacity. The shear capacity of the
concrete is estimated by equation 4-17 following the ACI recommendation for flexure
shear cracking of a reinforced concrete beam. The shell is assumed to take any
additional shear above this level.
V v A f
v MPa unitsv psi units
c k c
k
k
=
≤ ≤≤ ≤
'
. .. .
016 0 2919 35
(4-17)
For the second case the section is analyzed as monolithic, assuming no slip
occurs between the concrete core and the shell. A plane-sections-remain-plane
assumption is carried throughout thus assuming that the shear deformations are
insignificant compared to the bending deformations. Referring to Figure 4-9 at a
given angle θ the total load above the section of interest can be calculated. At a
distance dx along the member the moment has changed by an amount
M M Vdx' = + . (4-18)
For a small value of dx the neutral axis location is assumed constant and the total load
above the section of interest at a distance dx from the initial section is given by
P P M VdxM
'= + . (4-19)
53
The shear that must be transferred across the shear reference plane is thus
q P Pdx
PVM
= − =' . (4-20)
From the experimental data it can be seen that the shear strain in the shell increases
faster than the applied shear. This is attributed to the reduced shear carrying capability
of the concrete core. An effective thickness is used to calculate the shear strain from
the shear flow. The thickness assigned to the concrete is reduced by a ratio of the
mean concrete modulus above the shear reference plane to the modulus of the shell in
the longitudinal direction.
t t zEEeff
ca
L
= +2 , (4-21)
where the average concrete modulus is calculated from
EE dy zdy zcac= �
�. (4-22)
The concrete tangent modulus Ec is defined by equation (4-12).
θ
cy
z
N.A.
Figure 4-9 Geometric Properties for Determination of Shear Stress
54
4.1.4 Bending
A moment curvature program was written based on the above relations for
compression, tension and shear behavior. The maximum compression strain is
increased incrementally. The program iterates on the neutral axis location until
equilibrium is achieved. The plies in the shell are checked at each step until the first
fiber failure is encountered. This analysis can be entered with the constitutive fiber
and matrix properties that are then used to calculate lamina properties or the lamina
properties can be entered directly. Figure 4-10 maps out the analysis flow to
determine the bending capacity.
FIBER PROPERTIES
MATRIX PROPERTIESHAHN�S EQUATIONS
LAMINA PROPERTIES
CLASSICAL LAMINATION THEORY
EQUIVALENT HOMOGENEOUSORTHOTROPIC PLATE PROPERTIES.
CONCRETE PROPERTIES
CONCRETE 3-D COMPRESSIONCONSTITUTIVE EQUATIONS
LINEAR ELASTIC 2-DCONSTITUTIVE EQUATIONS
INCREMENTAL DEVELOPMENTOF COMPRESSION BEHAVIORFOR CYLINDRICAL SHELLSFILLED WITH CONCRETE
FIRST PLY FAILURE CRITERIA
TENSION BEHAVIORFOR CYLINDRICAL SHELLSFILLED WITH CONCRETE MOMENT CURVATURE ANALYSIS BENDING CAPACITY
Figure 4-10 Analysis Flow For Bending Behavior
55
The moment curvature program is based on conventional reinforced concrete
analysis with the main difference being the importance of accurately predicting the
hoop and shear strains in the shell throughout the loading so that the complete biaxial
stress state of the shell is known and can be checked against the ply allowables.
Classic lamination theory as described in Chapter 3 is used to determine the equivalent
orthotropic plate properties for the shell. To initialize the program an increment of
compression strain is applied to the extreme compression fiber of the member, a
neutral axis location is assumed for the composite system of shell and concrete core
and then the shell and core are integrated to determine the load and moment balance of
the section. The shell behavior can be determined from the biaxial linear elastic
equations presented in equations 4-1 and 4-2. Given the axial strain, the radial strain
can then be determined based on the appropriate compression or tension relation.
Utilizing the axial and radial strains the axial stress is then established. The total load
and moment in the shell are determined by integrating around the shell in 1o
increments from 00-180o and multiplying this result by 2. The concrete is divided into
horizontal slices or layers. The radial strain is determined based on the axial strain in a
given slice. The strain state is then used to estimate the axial stress in the slice. The
neutral axis location is iterated until equilibrium is achieved. For each increment of
compression strain applied the shell is checked for first ply failure. This is
accomplished by once again stepping around the shell in 1o increments calculating the
axial, radial and shear strains in the shell in the structure coordinate system,
transforming these strains to the ply coordinate system for each ply angle (since no
56
bending is assumed to exist in the shell each ply angle need only be checked once
regardless of the number of plies in the laminate at this angle), calculating the ply
stresses and comparing these ply stresses to the ply allowables. The analysis is
continued until a fiber or shear failure is encountered. The program output includes the
equivalent orthotropic plate properties calculated for the shell, the moment at first
matrix cracking and the moment and secant stiffness at first ply failure.
The concrete contribution in tension is generally considered to be negligible.
As described in Chapter 8 the effects of considering tension stiffening in the concrete
for a beam in bending are very small although if a beam and slab system is being
considered where the entire concrete filled fiber reinforced shell may be in tension
these effects could begin to be considerable.
4.2 Conrec Shells
A second cross section geometry considered in this document is a square
section with rounded corners that is referred to as a conrec section (see Figure 4-11).
The analysis for the conrec sections is based on the relations developed for the circular
sections. The area strain is used to calculate the concrete tangent modulus just as in the
case of the circular section. Finite element models have been used to investigate the
area strain patterns in the conrec sections as well as to quantify the effect of local shell
bending. Parameter studies were done for shells with 90o and ±10o plies. The ±10o
plies were increased from 0% of the lay-up to 80% of the lay-up. Studies were also
performed with various flat to radius ratios ranging from .5 to 3.
57
RadiusFlat
Figure 4-11 Conrec Cross Section
4.2.1 Compression
As mentioned in Section 4.1.1.2 the area strain in a circular section under
uniform compression is constant across the section. The models developed for the
circular cross section take advantage of this trait in that only one value of the
maximum Poisson�s ratio is necessary to characterize the behavior of the entire
section. When we get away from the circular cross section the area strain will vary
around the cross section and each point will experience different behavior based on
how much confinement the shell offers in that specific area of the section. An
incremental finite element approach has been developed that allows each element in
the model to respond differently under an applied uniform compression load. A small
increment of compression strain is applied to the models, the area strain in the
concrete core throughout the section, the hydrostatic pressure, as well as the hoop
strain in the shell are recorded. The area strain at the centroid of each element is used
to calculate a new Young�s modulus for that element. As explained in Section 4.1.1.2
the maximum equivalent tangent Poisson�s ratio occurs at a radial strain of 0.2% for
the circular sections. This is equivalent to an area strain of 0.4% (εA=2εr). The
58
hydrostatic pressure is used along with a linear relation (equation 4-27) to predict the
hydrostatic pressure that would be expected at an area strain of 0.4%.
σσ
εhyd mhyd
A_
.=
0 004 (4-27)
The predicted hydrostatic pressure is then used to estimate the maximum equivalent
tangent Poisson�s ratio from which the current equivalent tangent Poisson�s ratio is
calculated as described for the circular section. This process is repeated for subsequent
strain steps. The strains and stresses are accumulated for all steps until the final axial
strain of interest has been achieved.
To investigate the behavior of the conrec sections under compression finite
element modeling was used to determine the effect of varying the shell lay-up and the
flat to radius ratio. The models were constructed with one layer of solid elements to
model the concrete and composite shell elements around the perimeter (Figure 4-12)
to model the composite shell with proper bending stiffness (equivalent plate properties
described in Section 3.1.4 are for in-plane properties only). Elastic properties were
assumed for the concrete core. One quarter of the section was modeled with symmetry
boundary conditions being used. An equivalent circular section was also analyzed for
each lay-up with the same composite shell thickness and a diameter equal to the flat to
flat distance for the conrec. The area strains and hoop strains from the conrec sections
were ratioed to the values determined from the �equivalent circular sections�.
59
Figure 4-13 shows the 4 flat to radius ratios studied in this analysis. The flat to
flat dimension for all models was held constant at 381mm (15in.). A symmetric lay-up
of the form shown in Table 4-2 was assumed for all cases with the ply thicknesses
varied to give the desired percentage of helical plies. Three dimensional
representations of the area strain ratio (conrec/circular) for the shells with all hoop
plies are shown in Figure 4-14 through Figure 4-17 to demonstrate the effect of the
conrec geometry on the confinement of the concrete core. These plots represent one
quarter of the conrec section as shown in Figure 4-12 with the center at (0,0). These
plots look the same for all lay-ups studied with the exception of the scale of the strain
ratio axis. In Figure 4-18 through Figure 4-20 the values of the strain ratio for three
points in the section are given for the different shell geometries and lay-ups.
x y
z
y-symmetry x-symmetry
Figure 4-12 Finite Element Model Used for Evaluation of Conrec Sections
60
It can be seen from these plots that the shells with high hoop stiffness (high
percentage of 90o fibers) show a much greater loss of confinement efficiency (higher
area strains) than do the shells with low hoop stiffness (high percentage of ±10o
fibers). This is due to the fact that the circular sections with low hoop stiffness are not
very effective at confining the concrete core due to the high Poisson�s ratio of the shell
for loading in the longitudinal direction for these lay-ups. Bending applications for
which the shells with high axial stiffness are most advantageous do not suffer much
from the confinement losses as is demonstrated in the following section. For
compression behavior the shells with predominantly hoop fibers excel. The mean area
strain is an indicator of the response of the member in compression as this is a direct
indication of the load carrying capacity. The sections with large radii (F/R≤1) maintain
an area strain approximately 10% greater than that of the equivalent circular section
for an all hoop shell. As the flat to radius ratio increases the area strain as compared to
the equivalent circular section also increases. From this study a flat to radius ratio of 3
with a shell composed of all hoop fibers showed a mean area strain increase of
approximately 20% over the circular section.
61
Table 4-2 Composite Lay-Ups Used for Conrec Studies
% ±10o Plies Lay-up Ply Thickness mm (in.)
0 [90] [10.2]
([0.40])
50 [90,10,-10]sym [2.54,1.27,1.27]sym
([0.10,0.05,0.05]sym)
80 [90,10,-10]sym [1.02,2.03,2.03]sym
([0.04,0.08,0.08]sym)
a) 1/2 b) 1/1
c) 2/1 d) 3/1
x
y
x
y
x
y
x
yFlat/Radius
Figure 4-13 Conrec Geometries Used for This Analysis
62
0.90
0.95
1.00
1.05
1.10
1.15
1.20
1.25
01
23
45
67
1 2 3 4 56
7
Area S
train Ratio (C
onrec/Circ)
x
y
Profile of Section in x-y Plane
Figure 4-14 Area Strain Ratio Profile, 0% ±10o Plies, Flat to Radius Ratio .5
0.8
0.9
1.0
1.1
1.2
1.3
1.4
01
23
45
67
1 2 34 5
67
Area S
train Ratio (C
onrec/Circ)
x
y
Profile of Section in x-y Plane
Figure 4-15 Area Strain Ratio Profile, 0% ±10o Fibers, Flat to Radius Ratio of 1
63
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
01
23
45
67
1 2 3 45
67
Area S
train Ratio (C
onrec/Circ)
x
y
Profile of Section in x-y Plane
Figure 4-16 Area Strain Ratio Profile, 0% ±10o Fibers, Flat to Radius Ratio of 2
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
01
23
45
67
1 2 34 5
67
Area S
train Ratio (C
onrec/Circ)
x
y
Profile of Section in x-y Plane
Figure 4-17 Area Strain Ratio Profile, 0% ±10o Fibers, Flat to Radius Ratio of 3
64
Flat/Radius Ratio
0.5 1.0 1.5 2.0 2.5 3.0
Are
a S
trai
n R
atio
(C
on
rec/
Cir
cula
r)
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
Figure 4-18 Area Strain Ratio for 0% ±10o Conrec Shell
Flat/Radius Ratio
0.5 1.0 1.5 2.0 2.5 3.0
Are
a S
trai
n R
atio
(C
on
rec/
Cir
cula
r)
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
Figure 4-19 Area Strain Ratio for 50% ±10o Conrec Shell
65
Flat/Radius Ratio
0.5 1.0 1.5 2.0 2.5 3.0
Are
a S
trai
n R
atio
(C
on
rec/
Cir
cula
r)
0.94
0.96
0.98
1.00
1.02
1.04
Figure 4-20 Area Strain Ratio for 80% ±10o Conrec Shell
Unlike in the circular sections the hoop stresses in the conrec shells vary
around the section and also through the thickness due to bending. Variation of the
hoop stress is dependent on the shell lay-up and the geometry of the shell. Figure 4-21
through Figure 4-24 show the variation in the hoop stresses extracted from the finite
element models for the inside and outside surface of the shell ratioed to that of the
equivalent circular section at the same axial strain. The horizontal axis represents one
quarter of the perimeter of the shell starting from the center of a flat side as shown in
the figures. It can be seen from these plots that the maximum tension strain occurs in
the shell at the center of the flat on the outside surface for the geometries with a large
radius but can occur near the start of the radius due to bending if the radius is small.
66
Distance Around Section (mm)
0 50 100 150 200 250 300
Str
ess
Rat
io (
con
rec/
circ
)
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.20
Distance Around Section (in)
0 2 4 6 8 10 12
Outside 0% ±10o
Inside 0% ±100
Outside 50% ±10o
Inside 50% ±10o
Outside 80% ±10o
Inside 80% ±10o
Start of Radius
Area for Hoop Stress Plots
Figure 4-21 Hoop Stress in Shells With Flat To Radius Ratio of .5
Distance Around Section (mm)
0 50 100 150 200 250 300
Str
ess
Rat
io (
con
rec/
circ
)
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
Distance Around Section (in)
0 2 4 6 8 10 12
Outside 0% ±10o
Inside 0% ±10o
Outside 50% ±10o
Inside 50% ±10o
Outside 80% ±10o
Inside 80% ±10o
Area for Hoop Stress Plots
Start ofRadius
Figure 4-22 Hoop Stress in Shells With Flat To Radius Ratio of 1
67
Distance Around Section (mm)
0 50 100 150 200 250 300
Str
ess
Rat
io (
con
rec/
circ
)
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Distance Around Section (in)
0 2 4 6 8 10 12
Outside 0% ±10o
Inside 0% ±10o
Outside 50% ±10o
Inside 50% ±10o
Outside 80% ±10o
Inside 80% ±10o
Area for Hoop Stress Plots
Start ofRadius
Figure 4-23 Hoop Stress in Shells With Flat To Radius Ratio of 2
Distance Around Section (mm)
0 50 100 150 200 250 300
Str
ess
Rat
io (
con
rec/
circ
)
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Distance Around Section (in)
0 2 4 6 8 10 12
Outside 0% ±10o
Inside 0% ±10o
Outside 50% ±10o
Inside 50% ±10o
Outside 80% ±10o
Inside 80% ±10o
Area for Hoop Stress Plots
Start ofRadius
Figure 4-24 Hoop Stress in Shells With Flat To Radius Ratio of 3
68
4.2.2 Bending
The bending analysis for the conrec sections is carried out exactly as for the
circular sections. The concrete stress strain relation is developed using an equivalent
circular section which is a circular section with the same lay-up as the conrec section
and a radius equal to the flat to flat distance of the conrec. The following analysis is
presented to show that the use of the equivalent circular section for the bending
calculations does not effect the system response. The contribution to the bending
stiffness from the concrete is greatest for shells with predominantly hoop fibers. This
analysis was carried out for a conrec shell with a 127mm (5 in.) flat and a 127mm (5
in.) radius. The shell was carbon with the ply properties presented in Table 5-2, and
had a total thickness of 10.2mm (0.4 in.) with 90% of the fibers in the hoop direction
and 10% in the ±10o direction. The concrete strength assumed was 45.5 MPa (6.6 ksi).
Figure 4-25 shows the moment curvature response for this section and for the shell
alone. The difference between these two curves is the contribution to the behavior
provided by the concrete core. To show that the concrete model used has no great
effect on the overall behavior two concrete models were assumed, one based on the
equivalent circular section and one based on a perfectly plastic behavior cutting off the
concrete strength at f�c. This second model is a lower limit on the concrete strength.
Figure 4-26 shows the difference in the concrete stress strain behavior assumed.
When these models are used in a moment curvature analysis no discernible difference
is seen between the two (see Figure 4-27). This is due to the fact that the ultimate
strain for the shell is approximately 1.2% in the longitudinal direction which only puts
69
a small amount of concrete in a range where the choice of concrete model makes a
difference. For the parametric studies presented in Chapter 7 the equivalent circular
section is used to predict the behavior of the concrete in compression. If very low shell
thicknesses are being investigated it may become necessary to take into account the
variation in the area strain of the conrec section when performing a bending analysis.
Experimental investigations carried out for validation and calibration of the
models presented in this chapter will be discussed in the following sections.
Curvature (1/in)
0.0000 0.0004 0.0008 0.0012
Mo
men
t (k
ip-i
n)
0.0e+0
1.0e+6
2.0e+6
3.0e+6
Curvature (1/mm)
0e+0 1e-5 2e-5 3e-5 4e-5 5e-5
Mo
men
t (k
N-m
)
0.0e+0
1.0e+5
2.0e+5
3.0e+5
4.0e+5
Full SectionShell alone
Figure 4-25 Moment Curvature of Typical Conrec Section
70
Strain
-0.015-0.010-0.0050.000
Str
ess/
f'c
0.0
0.5
1.0
1.5
Equivalent Circular SectionPerfectly Plastic
Figure 4-26 Concrete Stress Strain Relation
Curvature (1/in)
0.0000 0.0004 0.0008 0.0012
Mo
men
t (k
ip-i
n)
0.0e+0
1.0e+6
2.0e+6
3.0e+6
4.0e+6
Curvature (1/mm)
0e+0 1e-5 2e-5 3e-5 4e-5 5e-5
Mo
men
t (k
N-m
)
0.0e+0
1.0e+5
2.0e+5
3.0e+5
4.0e+5
5.0e+5Cicular Relations for Concrete BehaviorPerfectly Plastic Relation for Concrete Behavior
Figure 4-27 Comparative Moment Curvature for Conrec Section With Various
Concrete Models
71
5. EXPERIMENTAL PROGRAM TO VALIDATE CONCRETE FILLED FRP TUBE BEHAVIOR
An experimental program designed to validate the concept of using concrete
filled FRP shells for bridge components has been under way at UCSD for the past
several years. Bridge column studies comparing concrete filled carbon reinforced
shells with various connection details to a standard reinforced concrete column have
been completed [38][39]. This document outlines: (1) compression tests done on small
scale concrete filled carbon fiber reinforced composite cylinders to investigate the
behavior of the concrete confined by a linear elastic shell, (2) small and full scale
bending tests to validate the models described in Chapter 4, and (3) a beam-and-slab
assembly with a cast in place concrete deck used here to investigate the stress
concentrations around the penetrations for the deck to beam connection and elsewhere
for evaluation of the system characterization [3][5]. Currently, a full scale bridge
section with concrete filled carbon girders and E-Glass/polyester deck panels is being
cyclically loaded to verify the fatigue response of such a proposed system (see Figure
1-3).
5.1 Small Scale Shells
The small scale test program consisted of bending, compression and thermal
expansion tests. Two shapes were investigated, circular cylinders with a diameter of
152mm (6 in.) and �conrec� cylinders. The conrec is a 152mm (6 in.) square with
72
51mm (2 in.) radii at the corners as shown in Figure 5-1. These shells were filament
wound with carbon epoxy by Alliant Techsystems Inc. A total of eight 2.44m (8 ft)
shells were manufactured. One set of shells was wound with purely hoop fibers. These
shells were utilized for compression tests. Another set of shells had approximately
85% of the fibers at ±10o from the longitudinal axis and 15% hoop fibers. A 2.13m (7
ft) section from these specimens was used for the bending tests and the remaining
305mm (1 ft) was tested in compression. The shells used for the small scale test
program are summarized in Table 5-1. The vendor supplied ply properties are listed in
Table 5-2. These properties were used to generate the equivalent plate properties for
the shells as listed in Table 5-3.
152mm6.0 in
152m
m
6.0
in
R51mm2.0 in
Figure 5-1 Nominal Geometry of Conrec Section
73
Table 5-1 Small Scale Test Shells NOMINAL
THICKNESS SHAPE LAY-UP S.I. U.S.
CIRC [90] 2.29mm .09 in.
CIRC [90] 4.57mm .18 in.
CIRC [90,±10,±10,90] 2.29mm .09 in.
CIRC [90,±10,±10,90]SYM 4.57mm .18 in.
CONREC [90] 2.29mm .09 in.
CONREC [90] 4.57mm .18 in.
CONREC [90,±10,±10,90] 2.29mm .09 in.
CONREC [90,±10,±10,90]SYM 4.57mm .18 in.
Table 5-2 Vendor Supplied Ply Properties
S.I. U.S.
E11 121 GPa 17.5 Msi
E22 6.90 GPa 1.0 Msi
G12 4.83 GPa 0.70 Msi
G23 2.62 GPa 0.38 Msi
ν12 0.30 0.30
ν23 0.40 0.40
74
Table 5-3 Equivalent Plate Properties
[90] [90,±10,±10,90]
S.I. U.S. S.I. U.S.
EL 6.90 GPa 1.00 msi 101 GPa 14.7 msi
EH 121 GPa 17.5 msi 19.8 GPa 2.87 msi
GLH 4.83 GPa 0.70 msi 7.52 GPa 1.09 msi
νLH 0.3 0.3 0.24 0.24
5.1.1 Concrete Characterization
Three 152mm (6 in.) diameter 305mm (1 ft) long concrete cylinders were cast
when the small scale shells were filled. These cylinders were tested in compression
with four longitudinal and four hoop strain gages so the stiffness, strength and
Poisson�s ratio of the concrete could be determined. The concrete mix used was a 35
MPa (5 ksi) nominal mix with light weight aggregate. Figure 5-2 shows the stress
strain curves for the three cylinders tested. The strain is the average of the four
longitudinal gages and the stress is derived from the applied load. The initial Young�s
modulus is determined from the average of these curves between 0 and 500
microstrain and was found to be 20.1 GPa (2.99 msi). This value correlates well with
the ACI recommended relation with weight (w) correction given by equation 5-1
which gives 19.9 GPa (2.88 ksi) using the nominal strength and 22.8 GPa (3.3 ksi)
using the measured strength.
E w fc c= 33 1 5( ). ' (psi units) (5-1)
75
The ultimate compression strength for the mix was calculated from the average
ultimate strength of the three cylinders and was found to be 45.5 MPa (6.6 ksi) at 28
days. Figure 5-3 shows the hoop strain vs. the longitudinal strain for all three
cylinders. Fitting this data leads to an initial Poisson�s ratio of 0.2. The concrete
properties used for the analysis are summarized in Table 5-4.
Longitudinal Strain (micro-strain)
-3500-3000-2500-2000-1500-1000-5000
Str
ess
(MP
a)
0
10
20
30
40
50
Str
ess
(ksi
)
0
1
2
3
4
5
6
7cyl #1cyl #2cyl #3
Figure 5-2 Concrete Compression Stress Strain Relation
76
Longitudinal Strain (micro-strain)
-3500-3000-2500-2000-1500-1000-5000
Ho
op
Str
ain
(m
icro
-str
ain
)
0
500
1000
1500
2000
2500
3000
cyl #1cyl #2cyl #3
Figure 5-3 Hoop Strain vs. Longitudinal Strain for Concrete Cylinders Under
Uniaxial Compression
Table 5-4 Experimentally Derived Concrete Properties
S.I. U.S.f�c 45.5 MPa 6.6 ksiEco 20.1 GPa 2.99 msiεco 0.003 0.003νco 0.2 0.2
5.1.2 Compression
The purely hoop shells were cut into three 610mm (2 ft) and one 305mm (1 ft)
sections for compression testing. One 305mm (1 ft) section was also cut from the
tubes with both hoop and helical fibers to be tested in compression. The matrix of test
77
specimens used for compression behavior characterization is shown in Table 5-5. It
was felt that an aspect ratio of 2 to 1 may not be sufficient to get data unpolluted by
the end effects. Finite element modeling of the circular cylinders was used to
investigate these end effects. Models with the ends free to expand radially were
compared with identical models with the ends fixed radially. This investigation
showed that the aspect ratio of the 610mm (2 ft) specimens, which was 4 to 1, should
give a zone in the center of the cylinder which is representative of an unrestrained
cylinder. An example of this analysis is shown below in Figure 5-4 for unconfined
circular cylinders with a Poisson�s ratio of 0.45. The typical test setup for the
compression tests is shown in Figure 5-5.
y/L
0.00 0.25 0.50 0.75 1.00
Ho
op
Str
ain
/ H
oo
p S
trai
n U
nre
stra
ined
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
UnrestrainedL/D=2/1L/D=4/1
Ly
D
Figure 5-4 Comparison of Hoop Strains for Cylinders With Various Aspect Ratios
78
Figure 5-5 Typical Compression Test Setup
79
Table 5-5 Compression Specimens
SHAPE LAY-UP LENGTH
THICKNESS
SPECIMEN DESIGNATION
CIRC [90] 610mm (2 ft)
2.3mm (0.09 in.)
c1,c2,c3
CIRC [90] 305mm (1 ft)
2.3mm (0.09 in.)
c4
CIRC [90] 610mm (2 ft)
4.6mm (0.18 in.)
c5,c6,c7
CIRC [90] 305mm (1 ft)
4.6mm (0.18 in.)
c8
CIRC [90,±10,±10,90] 305mm (1 ft)
2.3mm (0.09 in.)
c17
CIRC [90,±10,±10,90]sym 305mm (1 ft)
4.6mm (0.18 in.)
c18
CONREC [90] 610mm (2 ft)
2.3mm (0.09 in.)
c9,c10,c11
CONREC [90] 305mm (1 ft)
2.3mm (0.09 in.)
c12
CONREC [90] 610mm (2 ft)
4.6mm (0.18 in.)
c13,c14,c15
CONREC [90] 305mm (1 ft)
4.6mm (0.18 in.)
c16
CONREC [90,±10,±10,90] 305mm (1 ft)
2.3mm (0.09 in.)
c19
CONREC [90,±10,±10,90]sym 305mm (1 ft)
4.6mm (0.18 in.)
c20
80
5.1.2.1 Circular Cylinders
The circular cylinders were instrumented with four longitudinal strain gages
and four hoop strain gages as shown in Figure 5-6. Three linear potentiometers were
positioned around the specimens to evaluate the longitudinal strain at specimen
deformations past the capability of the strain gages.
The critical information derived from these tests is the stress vs. strain behavior
of the concrete confined by the linear elastic shell. An understanding of how the
concrete expands radially is essential to model this behavior. The load vs. longitudinal
strain plots for all four shell types tested are shown in Figure 5-7. It can be seen from
these plots that the thick all hoop shell offers the highest load carrying capacity. The
shells with the helical fibers are capable of carrying much more load in the axial
direction than the all hoop shells as the modulus of the helical shells is almost 15 times
greater than the all hoop shells in the loading direction. This explains why the thick
helical shell response looks very similar to the thin all hoop shell response as the
capacity lost in the concrete core due to more rapid expansion is made up by the load
carrying capability of the shell. The models presented in Chapter 4 base the concrete
modulus on the current expansion or damage state of the concrete. Figure 5-8 shows
the hoop strain vs. the longitudinal strain for the four shell architectures tested. As
expected the all hoop shells do a much better job of restraining the concrete. A
problem was encountered with the helical cylinders that rendered the data collected
from these tests very difficult to use. The hydrostone end caps broke away from the
shell fairly early on in the loading due to the high stiffness of the shells in the
81
longitudinal direction mentioned above. Once the end caps were gone the shells were
then free to expand longitudinally. The longitudinal strain in the specimen as
measured by the linear potentiometers is plotted along with the longitudinal strain as
recorded by the strain gages on the shell in Figure 5-9. Examination of these plots
leads to the conclusion that the desired state of equal longitudinal strain in the shell
and core was lost and that slippage was occurring between the two. A free edge
condition was created at the top and bottom of the shell meaning that any load in the
shell had to be transferred from the concrete core. Insufficient information is available
to assertion how this load transfer took place so these tests were not used in the
development of the analytical models. The stress strain response of the concrete core
is shown in Figure 5-10 for the all hoop shells. These curves are derived by estimating
the axial load in the shells and subtracting it from the total applied load to obtain the
load carried by the concrete core. An elastic perfectly plastic assumption was made for
the all hoop shells in the longitudinal direction with a yield strain of 24.8 MPa (3.6
ksi). The failure of the all hoop cylinders was dramatic with audible fiber failure and
substantial discoloration of the matrix observed before final failure. The hoop strains
at failure were in the range of 0.7% to 0.8%. The thin and thick helical cylinders
demonstrated different failure modes with the thin shell splitting at a 10o angle from
the vertical and the thick shell splitting vertically. Typical shell failures are depicted in
Figure 5-11 and Figure 5-12 for the all hoop and helical cylinders respectively.
82
HYDRASTONE END CAPS
BIAXIAL GAGES FOR LONGITUDINAL ANDHOOP STRAINS
610mm(2�)
152mm(6�)
B
B
B
B
B - BIAXIAL-LONGITUDINAL, HOOPLinearPots
- LINEAR POTS
Figure 5-6 Strain Gage Layout for Circular Compression Specimens
Longitudinal Strain (micro-strain)
-60000-50000-40000-30000-20000-100000
Lo
ad (
kN)
-3000
-2500
-2000
-1500
-1000
-500
0
Lo
ad (
kip
s)
-600
-500
-400
-300
-200
-100
0
Thin All Hoop ShellThick All Hoop ShellThin Shell With HelicalsThick Shell With Helicals
Figure 5-7 Load vs. Strain Curves for Circular Cylinders
83
Longitudinal Strain (micro-strain)
-60000-50000-40000-30000-20000-100000
Ho
op
Str
ain
(m
icro
-str
ain
)
0
2000
4000
6000
8000
Thin All HoopThick All HoopThin With HelicalsThick With Helicals
Figure 5-8 Hoop vs. Longitudinal Strains for Circular Cylinders
Load (kN)
-2000-1600-1200-800-4000
Lo
ng
itu
din
al S
trai
n (
mic
ro-s
trai
n)
-20000
-15000
-10000
-5000
0
Load (kips)
-400-350-300-250-200-150-100-500
Thin Shell Linear PotsThin Shell Strain GagesThick Shell Linear PotsThick Shell Strain Gages
Figure 5-9 Longitudinal Strain in Helical Circular Cylinders
84
Longitudinal Strain (micro-strain)
-40000-30000-20000-100000
Str
ess/
f'c
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Thin All Hoop ShellThick All Hoop Shell
Figure 5-10 Concrete Stress Strain Curves for All Hoop Circular Cylinders
Figure 5-11 Typical Failure of All Hoop Circular Shell
85
Figure 5-12 Failure of Helical Circular Shells
5.1.2.2 Conrec Cylinders
In order to capture the nonconstant hoop strains as described in Chapter 4 a
more complex strain gage layout was required for the conrec specimens as compared
to the circular specimens. Four biaxial gages were used to track the strains in the
center of the flat sides and uniaxial hoop gages were placed as shown in Figure 5-13
to track the varying hoop strains around the corner radius. The outside gages are
placed at the tangent points where the radius meets the flat and the center gage is
placed at the center of the radius.
86
For the conrec specimens the load vs. longitudinal strain behavior is depicted
in Figure 5-14. The pronounced kink in these curves for the conrec sections is
attributed to the rapid expansion of the concrete core that takes place due to the poor
confinement offered by the flat sides. This effect is mitigated as the flat bulges and
begins to better confine the section. The hoop strains in the shells are plotted vs. the
longitudinal strains in Figure 5-15 for the center of the flat side. This plot
demonstrates the rapid expansion in the initial stages of loading. The same problem
with the end-caps was encountered with the conrec helical cylinders as was described
above for the circular sections. Failure of the conrec cylinders was very similar to that
of the circular sections with hoop strains in the 0.4-1.0% range. Typical failures are
shown for the conrec sections in Figure 5-16 and Figure 5-17 for the all hoop and
helical cylinders respectively.
HYDRASTONE END CAPS
GAGES FOR HOOP STRAIN
BIAXIAL GAGES FOR LONGITUDINAL ANDHOOP STRAINS
610mm(2�)
152mm(6�)
B
B
B
B
HH
H
HH
H
B - BIAXIAL, LONGITUDINAL, HOOPH - HOOP
LinearPots
- LINEAR POTS
Figure 5-13 Strain Gage Layout for Conrec Compression Specimens
87
Longitudinal Strain (micro-strain)
-60000-50000-40000-30000-20000-100000
Lo
ad (
kN)
-3000
-2500
-2000
-1500
-1000
-500
0
Lo
ad (
kip
s)
-600
-500
-400
-300
-200
-100
0
Thin All HoopThick All HoopThin With HelicalsThick With Helicals
Figure 5-14 Load vs. Longitudinal Strain Conrec Cylinders
Longitudinal Strain (micro-strain)
-60000-50000-40000-30000-20000-100000
Ho
op
Str
ain
(m
icro
-str
ain
)
0
2000
4000
6000
8000
10000
12000
Thin All HoopThick All HoopThin With HelicalsThick With Helicals
Gage Location
Figure 5-15 Hoop vs. Longitudinal Strain for Conrec Cylinders
88
Figure 5-16 Typical Failure of All Hoop Conrec Shell
Figure 5-17 Failure of Conrec Helical Shells
89
5.1.3 Bending
Four point bending tests were performed on 2.13m (7 ft) long beams with the
shells listed in Table 5-6. The test setup is shown schematically in Figure 5-18 and
pictured in Figure 5-19. These tests were used to verify the bending behavior
postulated by the analytical models described in Chapter 4. Both circular and conrec
sections were tested. The four point bending test was chosen to give a constant
moment region in the center of the beam and a shear span on the two ends. The load
spacing was chosen to avoid a shear failure outside the constant moment region so the
bending capacity could be determined. The load was applied through 51mm (2 in.)
thick elastomeric pads. This load application method was chosen to distribute the load
and help to prevent a failure due to localized effects. The ends of the beam were
supported on pivots that could not introduce any axial load as they were free to slide in
the longitudinal direction.
Table 5-6 Shells for Small Scale Bending Tests
THICKNESS
SHAPE LAY-UP S.I. U.S.
CIRC [90,±10,±10,90] 2.29mm .09 in.
CIRC [90,±10,±10,90]SYM 4.57mm .18 in.
CONREC [90,±10,±10,90] 2.29mm .09 in.
CONREC [90,±10,±10,90]SYM 4.57mm .18 in.
90
1.83m(6�)
152mm(6�)
406mm (16�)
711mm (28�)
Shear SpanConstant Moment
Region
Figure 5-18 Schematic of Four Point Bending Test Setup
Figure 5-19 Four Point Bending Test on Small Scale Specimen
91
5.1.3.1 Circular Cylinders
The circular bending specimens were instrumented with strain gages to verify
the hoop to longitudinal strain relations and the shear behavior put forth in Chapter 4.
The strain gage layout used is presented in Figure 5-20.
A
A
T T
B
B
T
T
B
B
B
B
B
C
C
B
B
D
D
A-A B-B C-C D-D
12
45 6
7
8
3
L
L
BL
L
457mm18 in
559mm
22 in
51mm2 in
610mm24 in
T - Triaxial - Longitudinal, +45, -45B - Biaxial - Longitudinal, HoopL - Longitudinal
Figure 5-20 Strain Gage Layout for Circular Bending Specimens
The thin circular shell reached a maximum moment of 44.3 kN-m (392 kip-
in.) with a peak center displacement of 48.0mm (1.89 in.) relating to a maximum
compression strain of -0.54% and a maximum tension Strain of 0.99%. The load vs.
center displacement of the specimen is shown in Figure 5-21. The response is nearly
linear. Figure 5-22 and Figure 5-23 show the strain profile across the section in the
constant moment region and shear span respectively. It can be seen from these figures
that the section did not deform as predicted by the plane-sections-remain-plane
assumption. Some of the data presented below indicates that the shell was too thin to
92
distribute the applied load and that extensive local cracking of the concrete under the
load application points led to unpredictable behavior. The analytical models predict a
negative hoop strain in the shell in the tension zone of the specimen which was not
observed for this test. The positive hoop strains in the tension zone may be explained
by the formation of compression struts from the load application points to the lower
section of the shell. The longitudinal strains in the specimen are plotted vs. the applied
moment in Figure 5-24. In these and all following figures in this section the locations
in the shear span are designated with solid symbols and the locations in the constant
moment region are designated with open symbols. These plots show that the
longitudinal strains in the shear span of the specimen for a given moment are higher
than the corresponding strains in the constant moment region which should not be the
case if the specimen behaved as predicted. The next figure (Figure 5-25) plots the
hoop strains in the shell versus the applied moment. It is notable that the hoop strains
in the shear span are all positive and much higher than those seen in the constant
moment region of the specimen. This additional positive hoop strain is believed to be
from the more extensive cracking of the concrete due to shear. The shear strain in the
shell is plotted vs. the applied shear in Figure 5-26. This figure demonstrates the shear
behavior discussed in Section 4.1.3 with the shell initially showing little to no shear
strain and then suddenly increasing shear strains as the concrete reaches its cracking
strain. A pronounced drop in load was seen at a load of 108 kN (24 kips) after which
the specimen began to pick up load again. The specimen failed on the compression
side in the constant moment region.
93
Center Displacement (mm)
0 10 20 30 40
Lo
ad (
kN)
0
20
40
60
80
100
120
140
Center Displacement (in)
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75
Lo
ad (
kip
s)
0
5
10
15
20
25
30
Figure 5-21 Load - Displacement Curve for Thin Circular Bending Specimen
Axial Strain (micro-strain)
-6000 -4000 -2000 0 2000 4000 6000 8000
Dis
tan
ce F
rom
Cen
ter
(mm
)
-60
-40
-20
0
20
40
60
Dis
tan
ce F
rom
Cen
ter
(in
)
-3
-2
-1
0
1
2
3
25 kN50 kN75 kN100 kN
Load
25kN=5.62 kips
Figure 5-22 Strain Profile for Thin Circular Section in Constant Moment region
94
Dis
tan
ce f
rom
Cen
terl
ine
(mm
)
-60
-40
-20
0
20
40
60
Axial Strain (micro-strain)
-2000 -1000 0 1000 2000 3000 4000 5000
Dis
tan
ce f
rom
Cen
terl
ine
(in
)
-3
-2
-1
0
1
2
3
25kN50kN75kN100kN
Load
25kN=5.62 kips
Figure 5-23 Strain Profile for Thin Circular Section in Shear Area
Moment (kN-m)
0 5 10 15 20 25 30 35 40 45 50 55
Lo
ng
itu
din
al S
trai
n (
mic
ro-s
trai
n)
-8000
-6000
-4000
-2000
0
2000
4000
6000
8000
10000
12000
Moment (kip-in)
0 100 200 300 400 500
a1la2la3la5la6la7lc1lc2lc3lc5lc6lc7l
a c1 2
3
5
7
6
Figure 5-24 Longitudinal Strain vs. Moment For Thin Circular Specimen
95
Moment (kN-m)
0 5 10 15 20 25 30 35 40 45 50 55
Ho
op
Str
ain
(m
icro
-str
ain
)
-500
0
500
1000
1500
2000
2500
3000
Moment (kip-in)
0 50 100 150 200 250 300 350 400 450 500
a1ha2ha3ha5ha6ha7hc1hc2hc3hc5hc6hc7h
a c
1 2
3
5
7
6
Figure 5-25 Hoop Strain vs. Moment For Thin Circular Specimen
Shear (kN)
0 10 20 30 40 50 60 70
Sh
ear
Str
ain
(m
icro
-str
ain
)
-2000
0
2000
4000
6000
8000
10000
Shear (kips)
-4 -2 0 2 4 6 8 10 12 14
a2sa3sa6sa7s
a1 2
3
5
7
6
Figure 5-26 Shear Strain vs. Shear For Thin Circular Specimen
96
The thick circular section achieved a maximum moment of 102 kN-m (899
kip-in.) with the maximum tension and compression strains both in excess of 1.0%.
The load displacement behavior of the thick circular specimen is described in Figure
5-27. More nonlinear behavior is apparent in this specimen than in the thin shell. It can
be seen from Figure 5-28 and Figure 5-29 that for the thick shell the plane sections
remain plane assumption was justified. The longitudinal strains shown in Figure 5-30
behave as predicted in the shear span and in the constant moment region. The hoop
strains in the constant moment region shown in Figure 5-31 behaved as predicted by
the analytical models although the increased positive hoop strains seen in the shear
span are at this time unaccounted for in the modeling. The shear strains are shown in
Figure 5-32. The bilinear approximation does not match as well as for the thin shell
although it is still a fair prediction of the behavior (see Chapter 6). The failure of the
thick circular section is pictured in Figure 5-33.
97
Center Displacement (mm)
0 10 20 30 40 50
Lo
ad (
kN)
0
50
100
150
200
250
300
Center Displacement (in)
0.0 0.5 1.0 1.5
Lo
ad (
kip
s)
0
10
20
30
40
50
60
Figure 5-27 Load - Displacement Curve for Thick Circular Bending Specimen
Axial Strain (micro-strain)
-10000 -5000 0 5000 10000
Dis
tan
ce F
rom
Cen
ter
(mm
)
-60
-40
-20
0
20
40
60
Dis
tan
ce F
rom
Cen
ter
(in
)
-3
-2
-1
0
1
2
3
25 kN50 kN75 kN100 kN125 kN150 kN 175 kN200 kN225 kN250 kN
Load
25kN=5.62 kips
Figure 5-28 Strain Profile for Thick Circular Section in Constant Moment region
98
Axial Strain (micro-strain)
-4000 -2000 0 2000 4000 6000
Dis
tan
ce f
rom
Cen
terl
ine
(mm
)
-60
-40
-20
0
20
40
60
Dis
tan
ce f
rom
Cen
terl
ine
(in
)
-3
-2
-1
0
1
2
3
25kN50kN75kN100kN125kN150kN175kN200kN225kN250kN
Load
25kN=5.62 kips
Figure 5-29 Strain Profile for Thick Circular Section in Shear Span
Moment (kN-m)
0 10 20 30 40 50 60 70 80 90 100 110 120 130
Lo
ng
itu
din
al S
trai
n (
mic
ro-s
trai
n)
-15000
-12500
-10000
-7500
-5000
-2500
0
2500
5000
7500
10000
12500
15000
Moment (kip-in)
0 200 400 600 800 1000 1200
a1la2la3la5la6la7lc1lc2lc3lc5lc6lc7l
a c1 2
3
5
7
6
Figure 5-30 Longitudinal Strain vs. Moment For Thick Circular Section
99
Moment (kN-m)
0 10 20 30 40 50 60 70 80 90 100 110 120 130
Ho
op
Str
ain
(m
icro
-str
ain
)
-2000
-1000
0
1000
2000
3000
4000
Moment (kip-in)
0 200 400 600 800 1000 1200
a1ha2ha3ha5ha7hc1hc2hc3hc5hc6hc7h
a c1 2
3
5
7
6
Figure 5-31 Hoop Strain vs. Moment For Thick Circular Section
Shear (kN)
0 20 40 60 80 100 120 140 160
Sh
ear
Str
ain
(m
icro
-str
ain
)
-2000
0
2000
4000
6000
8000
10000
Shear (kips)
0 5 10 15 20 25 30 35
a2sa3sa7s
a1 2
3
5
7
6
Figure 5-32 Shear Strain vs. Applied Shear For Thick Circular Specimen
100
Figure 5-33 Failure of Thick Circular Section
5.1.3.2 Conrec Cylinders
The cross section geometry of the conrec section is shown in Figure 5-1. A
strain gage layout similar to that used for the circular bending specimens was used for
the conrec sections. Hoop gages were placed at the tangent points of the radius and at
the center of the radius in the constant moment region to characterize the hoop strain
distribution around the section. The strain gage layout for the conrec bending
specimens is shown in Figure 5-34.
The thin conrec section failed at a moment of 54.4 kN-m (482 kip-in.) with a
peak center displacement of 31.0mm (1.22 in.) (see Figure 5-35) corresponding to a
maximum compression strain of -0.56% and a maximum tension strain of 0.76%. The
101
A
A
C
C
B
B
D
D
A-A B-B C-C D-D
12
45
6 7 8
3 L
L910
B
B
T
T T
T
B
B
HHHB
H HHB
L
L
457mm18 in
559mm22 in
51mm2 in
610mm24 in
T - Triaxial - Longitudinal, +45, -45B - Biaxial - Longitudinal, HoopL - LongitudinalH - Hoop
Figure 5-34 Strain Gage Layout for Conrec Bending Specimens
section behaved well and showed no signs of section warpage as can be seen in Figure
5-36 and Figure 5-37. The longitudinal strains responded the same in the shear and
constant moment spans as shown in Figure 5-38. As was seen in the circular sections
the hoop strains in the shear span are higher and much more erratic than the
corresponding hoop strains in the constant moment region (see Figure 5-39), again
this is attributed to the additional dilation due to shear cracking. The hoop strain in the
center of the flat on the compression side of the member is seen to decrease at the
higher load levels. The shear strains exhibit the bilinear behavior postulated in Chapter
4 and as expected the peak shear strains are found near the centerline of the shell as
shown in Figure 5-40. The thin conrec specimen failed on the compression side in the
constant moment region as seen in Figure 5-41.
102
Center Displacement (mm)
0 5 10 15 20 25 30 35
Lo
ad (
kN)
0
20
40
60
80
100
120
140
160
180
Center Displacement (in)
0.0 0.5 1.0
Lo
ad (
kip
s)
0
5
10
15
20
25
30
35
40
Figure 5-35 Load - Displacement Curve for Thin Conrec Bending Specimen
Axial Strain (micro-strain)
-6000 -4000 -2000 0 2000 4000 6000 8000
Dis
tan
ce f
rom
Cen
terl
ine
(in
)
-3
-2
-1
0
1
2
3
Dis
tan
ce f
rom
Cen
terl
ine
(mm
)
-60
-40
-20
0
20
40
60
25kN50kN75kN100kN125kN150kN
Load
25kN=5.62
Figure 5-36 Strain Profile for Thin Conrec Section in Constant Moment region
103
Axial Strain (micro-strain)
-2000 -1000 0 1000 2000 3000 4000
Dis
tan
ce f
rom
Cen
terl
ine
(in
)
-3
-2
-1
0
1
2
3
Dis
tan
ce f
rom
Cen
terl
ine
(mm
)
-60
-40
-20
0
20
40
60
25kN50kN75kN100kN125kN150kN
Load
25kN=5.62 kips
Figure 5-37 Strain Profile For Thin Conrec Section in Shear Area
Moment (kN-m)
0 5 10 15 20 25 30 35 40 45 50 55 60 65
Lo
ng
itu
din
al S
trai
n (
mic
ro-s
trai
n)
-8000
-6000
-4000
-2000
0
2000
4000
6000
8000
10000
Moment (kip-in)
0 100 200 300 400 500 600
a1la3la5la6la8la10lc1lc5lc6lc10l
a c
1
5
6
10
234
98
7
Figure 5-38 Longitudinal Strain vs. Moment For Thin Conrec Specimen
104
Moment (kN-m)
0 5 10 15 20 25 30 35 40 45 50 55 60 65
Ho
op
Str
ain
(m
icro
-str
ain
)
-400
-200
0
200
400
600
800
1000
1200
1400
1600
1800
2000
Moment (kip-in)
0 100 200 300 400 500 600
a1ha3ha5ha6ha8ha10hc1hc2hc3hc4hc5h
a c1
5
6
10
234
98
7
Figure 5-39 Hoop Strain vs. Moment For Thin Conrec Specimen
Shear (kN)
0 20 40 60 80
Sh
ear
Str
ain
(m
icro
-str
ain
)
0
2000
4000
6000
8000
10000
12000
Shear (kips)
0 2 4 6 8 10 12 14 16 18 20
a3sa5sa8sa10s
a
1
5
6
10
234
98
7
Figure 5-40 Shear Strain vs. Applied Shear For Thin Conrec Specimen
105
Figure 5-41 Failure of Thin Conrec Section
The thick conrec section exhibited similar behavior to the thin conrec
specimen. The thick conrec section failed at a moment of 135 kN-m (1200 kip-in.)
corresponding to a maximum compression strain of -0.88% and a maximum tension
strain of 1.0%. A slight loss of load capacity was seen at 267 kN (60 kips) as shown in
Figure 5-42. The center displacement was greater than the capacity of the measuring
device which only had a clearance of approximately 38mm (1.5 in.). The strain
profiles presented in Figure 5-43 and Figure 5-44 show that the plane-sections-
remain-plane assumption held for this specimen. As with the previous tests the
longitudinal strains responded as expected (see Figure 5-45) and the hoop strains in
the shear span were much higher than those in the constant moment region (Figure
106
5-46). Again the bilinear nature of the shear response was observed as shown in
Figure 5-47. The thick conrec section as all of the other bending specimens failed in
the constant moment region on the compression side of the shell. The failed specimen
is pictured in Figure 5-48.
Center Displacement (mm)
0 10 20 30 40
Lo
ad (
kN)
0
100
200
300
Center Displacement (in)
0.0 0.5 1.0 1.5
Lo
ad (
kip
s)
0
20
40
60
Figure 5-42 Load-Displacement Response for Thick Conrec Specimen
107
Axial Strain (micro-strain)
-10000 -5000 0 5000 10000 15000
Dis
tan
ce f
rom
Cen
ter
(mm
)
-60
-40
-20
0
20
40
60
Dis
tan
ce F
rom
Cen
ter
(in
)
-3
-2
-1
0
1
2
3
25kN50kN75kN100kN125kN150kN175kN200kN225kN250kN275kN300kN325kN350kN375kN
Load
25kN=5.62kips
Figure 5-43 Strain Profile for Thick Conrec Section in Constant Moment region
Axial Strain (micro-strain)
-4000 -2000 0 2000 4000 6000
Dis
tan
ce f
rom
Cen
ter
(in
)
-3
-2
-1
0
1
2
3
Dis
tan
ce f
rom
Cen
ter
(mm
)
-60
-40
-20
0
20
40
60 25kN50kN75kN100kN125kN150kN175kN200kN225kN250kN275kN300kN325kN350kN375kN
Load
25kN=5.62kips
Figure 5-44 Strain Profile For Thick Conrec Section in Shear Area
108
Moment (kN-m)
0 20 40 60 80 100 120 140 160
Lo
ng
itu
din
al S
trai
n (
mic
ro-s
trai
n)
-10000
-5000
0
5000
10000
15000
Moment (kip-in)
0 200 400 600 800 1000 1200 1400
a1la3la5la6la8la10lc1lc5lc6lc10l
a c
1
5
6
10
234
98
7
Figure 5-45 Longitudinal Strain vs. Moment for Thick Conrec
Moment (kN-m)
0 20 40 60 80 100 120 140 160
Ho
op
Str
ain
(m
icro
-str
ain
)
-2000
-1000
0
1000
2000
3000
4000
Moment (kip-in)
0 200 400 600 800 1000 1200 1400
a1ha3ha5ha6ha8ha10hc1hc2hc3hc4hc7hc10h
a c
1
5
6
10
234
98
7
Figure 5-46 Hoop Strain vs. Moment for Thick Conrec Section
109
Shear (kN)
0 50 100 150 200
Sh
ear
Str
ain
(m
icro
-str
ain
)
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
Shear (kips)
0 10 20 30 40
a3sa5sa8sa10s
a
1
5
6
10
234
98
7
Figure 5-47 Shear Strain vs. Applied Shear for Thick Conrec Specimen
Figure 5-48 Failure of Thick Conrec Section
110
5.2 Full Scale Bending Tests
Five four point bending tests were conducted on 345mm (14 in.) diameter
shells. Two shells were tested hollow to determine the accuracy of the stiffness
predicted utilizing classic lamination theory as described in Chapter 3. Two shells
were also tested filled with concrete and one shell was tested filled, with an overlaid
cast in place concrete deck. The test setup is pictured in Figure 5-49 and shown
schematically in Figure 5-50.
Figure 5-49 Full Scale Four Point Bending Test
111
7.92m
2.44m
FIXED SUPPORT PINNED SUPPORTCARBON SHELL
ENDBLOCK
2.74m(26�0�)
(8�0�) (9�0�)
Figure 5-50 Schematic of Full Scale Bending Tests
The tests were designed to put the specimen in four point bending with no
introduction of axial load. This was accomplished by casting a steel reinforced
concrete end block onto each end of the shell with an inset pipe as shown in Figure 5-
51. The endblock to tube connection was accomplished with steel bars that penetrated
914mm (36 in.) into the shell from the end of the endblock. These endblocks were
then supported on pin supports one of which was free to rotate at its base. The pivoting
pin support allows for axial deformation which eliminates the introduction of axial
load into the test specimen.
112
Figure 5-51 Endblock for Support of Bending Test Specimens
The construction of these test specimens was done in a way to develop a
construction technique that could be duplicated in the field. The idea was to pump the
tubes full with lightweight concrete in the horizontal position. To accomplish this a
diaphragm was required to seal the end of the shell through which the concrete could
be pumped (Figure 5-52). This diaphragm supported the connection cage as well as
the inlet for pumping the shell. The construction sequence used on the test specimens
was as follows. The steel cage for the endblock to tube connection was tied with the
diaphragm placed on the bars to seal the end of the shell (Figure 5-53). This cage was
then fitted into the shell so the diaphragm was approximately 51mm (2 in.) into the
shell. A pin was inserted through small holes in the end of the shell to hold the
connection cage in place (Figure 5-54). The endblock was designed so that the main
113
part of the endblock steel cage was tied leaving a gap in the front so the shell could be
dropped straight in from above. This part of the cage for the endblock was assembled
and placed inside the endblock casting forms (Figure 5-55). The shell with the
connection cages in place was then drooped into the casting forms from above (Figure
5-56). The pipe for filling the shell was then placed through the endblock cage (Figure
5-57). The top of the endblock cage and forms were then completed (Figure 5-58).
The endblocks were cast with conventional concrete leaving the pipe protruding from
the top. After the endblock had sufficiently cured the shell was pumped full with
lightweight concrete.
Figure 5-52 Shell End Diaphragm
114
Figure 5-53 Steel Connection Cage
Figure 5-54 Shell With Connecting Cage
115
Figure 5-55 Endblock Lower Section
Figure 5-56 Placement of Shell Into Endblocks
116
Figure 5-57 PVC Pipe for Pumping Shell
Figure 5-58 Completed End Block Form
117
Two composite architectures were used for the hollow and filled tube tests.
The lay-ups used are listed in Table 5-7, the vendor supplied ply properties are given
in
Table 5-8 and the equivalent plate properties derived from lamination theory are listed
in Table 5-9
Table 5-7 Composite Architectures for Large Scale Tests
THICKNESS
# LAY-UP S.I. U.S.
1 [90,±102,90,±102,90]sym 9.65mm .38 in.
2 [902,±102,902,±102,902,±10,902,±10,903] 8.89mm .35 in.
Table 5-8 Vendor Supplied Ply Properties
S.I. U.S.
E11 121 GPa 17.5 Msi
E22 6.90 GPa 1.0 Msi
G12 4.83 GPa 0.70 Msi
G23 2.62 GPa 0.38 Msi
ν12 0.30 0.30
ν23 0.40 0.40
118
Table 5-9 Equivalent Plate Properties for Large Scale Tests
Lay-Up #1 Lay-Up #2
S.I. U.S. S.I. U.S.
EL 97.2 GPa 14.1 msi 80.67 GPa 11.7 msi
EH 25.1 GPa 3.64 msi 42.9 GPa 6.22 msi
GLH 7.38 GPa 1.07 msi 6.89 GPa 1.00 msi
νLH 0.184 0.184 0.097 0.097
5.2.1 Concrete Properties
The shells used in these tests were pumped full with a lightweight concrete
pumpmix. A man-made lightweight aggregate with a maximum size of 9.5mm (3/8
in.) supplied by Pacific Custom Materials was used along with an expansive agent
(Interplast-N) supplied by Sika Corporation. Finally plastisizers were added before
pumping to ensure that the mix would flow easily through the shells. Two separate
concrete batches were used for the large scale test program. The first batch was used
for the first filled shell test and the beam and slab test. The second batch was used for
the second filled shell test. The first batch did not achieve the nominal design strength
of 27.6 MPa (4 ksi). This was due to the fact that insufficient plastisizer was added
and additional water was used to get the mix to flow. Table 5-10 shows the concrete
mix used and the proportions of the additives. The measured properties of both
batches are given in Table 5-11.
119
Table 5-10 Concrete Mix Used for Filled Shells
Material Loose Volume
Batch Weight
(lb.)
Absolute Volume
(ft3) Cement, Type 2 7.5 sacks 705 3.59
Washed Concrete Sand, SSD - 1586 9.52 Baypor F-60, SSD @ 55.0 lb/ft3 13.0 ft3 715 7.39 Active Water 39.0 Gals. 325 5.21 Absorbed Water, Maximum 3.0 Gals. - - Total Water Allowed 42.0 Gals. - - WRDA 79 @ 5 fl. oz./cwt. - 32.9 fl. oz. - Daravair @ 0.5 fl. oz./cwt. 4% air 3.3 fl. oz. 1.08 Interplast-N @ 1.0% per cwt. 0.8% air 6.6 0.21 Plastisizer (ADVA) 3 fl. oz./cwt.
- 23 oz. -
*Target slump 8-9 in.
Table 5-11 Concrete Properties for Filled Shell Tests
Batch 1 Batch 2 f�c 20.7 MPa
(3.0 ksi) 31.0 MPa (4.5 ksi)
Ec
o
15.4 GPa (2.23 msi)
18.8 GPa (2.73 msi)
εco .003 .003
5.2.2 Hollow Shell
The hollow shell test specimens were fitted with endblocks the same as the
filled specimens except the location of the diaphragm was moved 762mm (30 in.) into
the shell to provide a sufficient bond length for the steel connection cage.
120
Two hollow tests were performed with composite architectures as listed in
Table 5-7. The structural stiffness was compared to analytical predictions based on the
individual ply properties. Due to the susceptibility of the unfilled shell to local stress
concentrations and bending effects (ovalization) at the load introduction points it was
decided to do nondestructive tests with loads chosen to avoid local failure.
Displacement readings were taken at 5 points along the top surface of the shell
and one point on the bottom surface. Rotation of the specimen was monitored with
rotational potentiometers at the south end. The location and designation of these
devices is shown in Figure 5-59. Strain gages were also placed on the specimen to
monitor the bending and shear strains in the shell. The location and designation of
these gages is also shown in Figure 5-59.
The load displacement curves for the two shells is given in Figure 5-60.
Figure 5-61 and Figure 5-62 show the strains at the maximum compression and
tension fibers in the constant moment region as well as the shear strain on the neutral
axis for both tests. All values correlated well with the predictions based on classic
lamination theory and the vendor supplied ply properties. The second shell was
substantially softer as expected. It is notable that the shear stiffness does not change
much between the two lay-ups as can be seen in Figure 5-62.
121
A
A
A-AB-B
B
B
B1
B2
A
1.22m(4'-0")
914mm(3'-0")
DESIGNATIONSSTRAIN GAGE
B1LLOCATION ORIENTATION
L-LONGITUDINALH-HOOPP- +45M- -45
1.22m(4'-0")
2.13m(7'-0")
T 1 T 2 T 3 T 4 T 5
B 2
610mm(2'-0")
2.44m(8'-0")610mm
(2'-0")
ROT_L
ROT_T
SUP 1 SUP 2
1.22m(4'-0")
1.22m(4'-0")
914mm(3'-0")
*ALL LINEAR POTS HAVE A DISP PREFIX (I.E. DISPT2)
Figure 5-59 Instrumentation Layout for Hollow Shell Test
Displacement (mm)
0 10 20 30 40 50
Lo
ad (
kN)
0
10
20
30
40
50
Displacement (in)
0.0 0.5 1.0 1.5
Lo
ad (
kip
s)
0
2
4
6
8
10Lay-Up #1Lay-Up #2
Figure 5-60 Load Displacement Curve for Hollow Shell Tests
122
Moment (kN-m)
0 20 40 60 80 100 120
Str
ain
(m
icro
-str
ain
)
-1500
-1000
-500
0
500
1000
1500
Moment (kip-in)
0 200 400 600 800 1000b1l_#1b1h_#1b2l_#1b2h_#1b1l_#2b1h_#2b2l_#2b2h_#2
b
1
2
Figure 5-61 Longitudinal and Hoop Strains for Hollow Shells
Shear (kN)
0 10 20 30 40 50
Sh
ear
Str
ain
(m
icro
-str
ain
)
0
200
400
600
800
1000
1200
Shear (kips)
0 2 4 6 8 10
Lay-Up #1Lay-Up #2
Gage Location
Figure 5-62 Shear Strains for Hollow Shell Tests
123
5.2.3 Concrete Filled Shells
Two carbon shells were tested filled with a 27.5 MPa (4 ksi) lightweight
concrete pump mix. These tests were used to verify the analytical modeling done to
predict the stress and strain state in the shell due to the bending loads along with the
expansion of the enclosed concrete. Two different composite lay-ups were used as
listed in Table 5-7. The ply properties are listed in
Table 5-8 and the equivalent plate properties are listed in Table 5-9.
Displacement readings were taken at 5 points along the top surface of the
shells. The location and designation of these linear potentiometers is shown in Figure
5-63 for lay-up #1 and Figure 5-59 for lay-up #2. Strain gages were also placed on the
specimens to monitor the bending and shear strains in the shell. The location and
naming convention of these gages is shown in Figure 5-64.
D1 D2 D3 D4 D5
1.22m(4'-0") (2'-8")
784mm 1.32m(4'-4")
R1 R2
SD1 SD2
610mm(2'-0")
610mm(2'-0")
784mm(2'-8")
1.32m(4'-4")
Figure 5-63 Displacement Instrumentation For Filled Shell Test
124
Although the girder alone is not representative of a beam and slab system a
load was chosen that roughly represented a service bending demand on the filled shells
(51.2 kN (11.5 kips) per actuator). The specimens were cycled three times at this load
to see if any degradation was noted in the system stiffness. None was observed on
either test unit. The specimens were then loaded to failure.
A
A C
C D
D
A-A
12
3
4
567
8
C-CB-B
B
B
D-D
1
2
1.52m(5'-0")
1.37m(4'-6")
457mm(1'-6") 1.83m
(6'-0")
DESIGNATION
LONGITUDINALLOCATION
A4P
RADIAL LOCATION
ORIENTATIONL - LONGITUDINALH - HOOPP - +45M - -45
Figure 5-64 Strain Gage Locations and Designation for Filled Tube Tests
5.2.3.1 Shell #1
The lay-up used for the first bending test is shown in Table 5-7. This lay-up
yields a radius to thickness ratio of 18.2 with 84% of the fibers in the helical direction.
Failure occurred on the compression side of shell #1 in the constant moment region at
a longitudinal compression strain of ~0.55%. This value is approximately 46% of the
125
theoretical allowable strain based on a first ply failure model (1.2%). The tension
strain at failure was approximately 0.75%. It is felt that the single layer of hoop fibers
on the exterior of the shell was insufficient to keep the ±10o fibers from buckling
outward. It must be noted however that once the carbon girder is integrated into the
beam and slab system the shell sees no compression loading under service or factored
loadings (see filled tube with slab test).
The load vs. displacement curve for the first filled shell test is presented in
Figure 5-65. The nonlinear behavior due to the concrete behavior is more apparent
than for the small scale specimens. The analytical models used to predict the shell
strains are based on a plane sections remain plane assumption. From Figure 5-66 and
Figure 5-67, which show the strain profile across the section in the constant moment
and shear spans of the specimen respectively, it can be seen that this assumption is
justified. Figure 5-68 and Figure 5-69 show the longitudinal strains from the
compression and tensions sides of the shell plotted against the moment in the
specimen at the specific gage location. The behavior of the longitudinal strains in the
two spans is essentially the same as would be expected. The hoop strains are shown
versus the moment in Figure 5-70 and Figure 5-71. Here as in the small scale bending
tests the increased hoop strains in the shear span are seen especially on the tension side
of the shell where the cracking of the concrete core would be most pronounced. Shear
strains in the shear span of the specimen are shown versus the applied shear in Figure
5-72. The maximum shear strains were seen close to the centerline of the shell. The
gradual increase of the shear strains from the initiation of the loading is consistent
126
with the shear model presented in Chapter 4 for a shell with sufficient shear transfer
mechanisms to force the shell and core to deform as a unit. The failed shell is pictured
in Figure 5-73.
Center Displacement (mm)
0 50 100 150 200 250
Lo
ad P
er A
ctu
ato
r (k
N)
0
50
100
150
200
250
300
Center Displacement (in)
0 2 4 6 8
Lo
ad P
er A
ctu
ato
r (k
ips)
0
10
20
30
40
50
60
Figure 5-65 Load Displacement Plot for Filled Shell Test #1
127
Strain (micro-strain)
-8000 -6000 -4000 -2000 0 2000 4000 6000 8000
Dis
tan
ce F
rom
Sh
ell C
ente
rlin
e (m
m)
-200
-150
-100
-50
0
50
100
150
200
Dis
tan
ce F
rom
Sh
ell C
ente
rlin
e (i
n)
-6
-4
-2
0
2
4
6
25 kN50 kN75 kN100 kN125 kN150 kN175 kN200 kN225 kN250 kN
Load
25kN=5.62 kips
Figure 5-66 Strain Profile in Constant Moment Section, Filled Shell Test #1
Strain (micro-strain)
-6000 -4000 -2000 0 2000 4000 6000
Dis
tan
ce F
rom
Sec
tio
n C
ente
rlin
e (i
n)
-6
-4
-2
0
2
4
6
Dis
tan
ce F
rom
Sec
tio
n C
ente
rlin
e (m
m)
-200
-150
-100
-50
0
50
100
150
200
25 kN50 kN75 kN100 kN125 kN150 kN175 kN200 kN225 kN250 kN
Load
25kN=5.62 kips
Figure 5-67 Strain Profile in Shear Section, Filled Shell Test #1
128
Moment (kN-m)
0 100 200 300 400 500 600 700
Lo
ng
itu
din
al S
trai
n (
mic
ro-s
trai
n)
-6000
-5000
-4000
-3000
-2000
-1000
0
Moment (kip-in)
0 1000 2000 3000 4000 5000 6000
a1a2a3c1c2c3
a c1 2
3
57
6
48
Figure 5-68 Longitudinal Strain vs. Moment in Comp. Zone For Filled Shell #1
Moment (kN-m)
0 100 200 300 400 500 600 700
Lo
ng
itu
din
al S
trai
n (
mic
ro-s
trai
n)
0
1000
2000
3000
4000
5000
6000
7000
Moment (kip-in)
0 1000 2000 3000 4000 5000 6000
a4a5a6a7a8c4c5c6c7c8
a c1 2
3
57
6
48
Figure 5-69 Longitudinal Strain vs. Moment in Tension Zone For Filled Shell #1
129
Moment (kN-m)
0 100 200 300 400 500 600 700
Ho
op
Str
ain
(m
icro
-str
ain
)
0
200
400
600
800
1000
1200
1400
Moment (kip-in)
0 1000 2000 3000 4000 5000 6000
a1a2a3c1c2c3
a c1 2
3
57
6
48
Figure 5-70 Hoop Strain vs. Moment in Compression Zone For Filled Shell #1
Moment (kN-m)
0 100 200 300 400 500 600 700
Ho
op
Str
ain
(m
icro
-str
ain
)
-600
-400
-200
0
200
400
600
Moment (kip-in)
0 1000 2000 3000 4000 5000 6000
a4a5a6a8c4c5c6c7c8
a c1 2
3
57
6
48
Figure 5-71 Hoop Strain vs. Moment in Tension Zone For Filled Shell #1
130
Shear (kN)
0 50 100 150 200 250
Sh
ear
Str
ain
(m
icro
-str
ain
)
0
500
1000
1500
2000
2500
3000
Shear (kips)
0 10 20 30 40 50
a2a3a4a6a8
a c1 2
3
57
6
48
Figure 5-72 Shear Strain vs. Applied Shear For Filled Shell #1
Figure 5-73 Failure of Filled Shell #1
131
5.2.3.2 Shell #2
The lay-up for the second shell is shown in Table 5-7. This lay-up yields a
radius to thickness ratio of 19.8 with 69% helical fibers. The lay-up used for the
second concrete filled shell bending test had more hoop fibers on the outer surface to
see if the buckling failure encountered in the first bending test could be mitigated. The
shell failed on the compression side in the constant moment region just inside one of
the actuators. The ultimate compression strain recorded was 0.83% which is a
substantial improvement over the 0.55% seen on the first shell. The load-displacement
curve for the second filled shell is shown in Figure 5-74. The ultimate center
displacement was 305mm (12 in.). The nonlinear response is more pronounced in this
specimen due to the increased contribution to the stiffness from the concrete core. The
two shells are compared along with the analytical predictions in Chapter 6. The strain
profiles across the section in both the constant moment and shear spans are not as
straight as for the shell in the first bending test as can be seen in Figure 5-75 and
Figure 5-76. The additional hoop fibers on the exterior of the shell produced a
relatively thick soft layer which may have contributed to the observed section
warpage. It also led to difficulties with the strain gages as is seen in the following
plots. The longitudinal strains on the compression and tension sides of the shell are
plotted vs. the moment in Figure 5-77 and Figure 5-78 respectively. The behavior
was as expected with the exception of the peak tension strain recorded in the constant
moment region which began to increase faster than expected in the later stages of
loading and eventually reached a strain of 1.4%. It is felt that this phenomena is
132
attributable to matrix cracking in the relatively soft layer on the outside of the shell in
the longitudinal direction (see Chapter 6). The hoop strain followed the trend of the
previous tests with the increased hoop strains once more observed in the shear span as
shown in Figure 5-79 and Figure 5-80 for the compression and tension sides of the
shell respectively. Figure 5-81 plots the shear strains vs. the applied shear load for the
second filled shell bending test. The maximum shear strains have moved away from
the centerline of the shell as compared to the first filled shell test. Again the response
is smooth and the behavior seems to correlate well with the second shear model
presented in Chapter 4. The compression failure of the shell is shown in Figure 5-82.
Center Displacement (mm)
0 50 100 150 200 250 300 350
Lo
ad P
er A
ctu
ato
r (k
N)
0
50
100
150
200
250
300
Center Displacement (in)
0 2 4 6 8 10 12
Lo
ad P
er A
ctu
ato
r (k
ips)
0
10
20
30
40
50
60
Figure 5-74 Load Displacement Plot for Filled Shell Test #2
133
Strain (micro-strain)
-10000 -5000 0 5000 10000 15000
Dis
tan
ce F
rom
Sec
tio
n C
ente
rlin
e (i
n)
-6
-4
-2
0
2
4
6
Dis
tan
ce F
rom
Sec
tio
n C
ente
rlin
e (m
m)
-200
-150
-100
-50
0
50
100
150
200
25 kN50 kN75 kN100 kN125 kN150 kN175 kN200 kN225 kN250 kN
Load
25kN=5.62 kips
Figure 5-75 Strain Profile in Constant Moment Section, Filled Shell Test #2
Strain (micro-strain)
-6000 -4000 -2000 0 2000 4000 6000 8000
Dis
tan
ce F
rom
Sh
ell C
ente
rlin
e (m
m)
-200
-150
-100
-50
0
50
100
150
200
Dis
tan
ce F
rom
Sh
ell C
ente
rlin
e (i
n)
-6
-4
-2
0
2
4
625 kN50 kN75 kN100 kN125 kN150 kN175 kN200 kN225 kN250 kN
Load
25kN=5.62 kips
Figure 5-76 Strain Profile in Shear Section, Filled Shell Test #2
134
Moment (kN-m)
0 100 200 300 400 500 600 700
Lo
ng
itu
din
al S
trai
n (
mic
ro-s
trai
n)
-9000
-8000
-7000
-6000
-5000
-4000
-3000
-2000
-1000
0
Moment (kip-in)
0 1000 2000 3000 4000 5000 6000
a1a2a3c1c2c3
a c1 2
3
57
6
48
Figure 5-77 Longitudinal Strain vs. Moment in Comp. Zone For Filled Shell #2
Moment (kN-m)
0 100 200 300 400 500 600 700
Lo
ng
itu
din
al S
trai
n (
mic
ro-s
trai
n)
0
1500
3000
4500
6000
7500
9000
10500
12000
13500
15000
Moment (kip-in)
0 1000 2000 3000 4000 5000 6000
a4a5a6a7a8c4c5c6c7c8
a c1 2
3
57
6
48
Figure 5-78 Longitudinal Strain vs. Moment in Tension Zone For Filled Shell #2
135
Moment (kN-m)
0 100 200 300 400 500 600 700
Ho
op
Str
ain
(m
icro
-str
ain
)
0
200
400
600
800
1000
1200
1400
1600
1800
Moment (kip-in)
0 1000 2000 3000 4000 5000 6000
a1a2a3c1c2c3
a c1 2
3
57
6
48
Figure 5-79 Hoop Strain vs. Moment in Compression Zone For Filled Shell #2
Moment (kN-m)
0 100 200 300 400 500 600 700
Ho
op
Str
ain
(m
icro
-str
ain
)
-800
-600
-400
-200
0
200
400
600
800
Moment (kip-in)
0 1000 2000 3000 4000 5000 6000
a4a5a6a7a8c4c5c6c7c8
a c1 2
3
57
6
48
Figure 5-80 Hoop Strain vs. Moment in Tension Zone For Filled Shell #2
136
Shear (kN)
0 50 100 150 200 250 300
Sh
ear
Str
ain
(m
icro
-str
ain
)
0
500
1000
1500
2000
2500
3000
3500
Shear (kips)
0 10 20 30 40 50 60
a2a3a4a6a7a8
a c1 2
3
57
6
48
Figure 5-81 Shear Strain vs. Applied Shear For Filled Shell #2
Figure 5-82 Failure of Filled Shell #2
137
5.2.4 Concrete Filled Shell with integral Concrete Deck
The final test in this series consisted of a carbon girder filled with lightweight
concrete overlaid by a normal weight concrete slab connected through steel shear
dowels. This test verified the bending stiffness of the beam and slab system [3] as well
as demonstrated the efficacy of the shear connections [5]. The carbon shell was
prepared identical to the filled beam tests with the exception of the placement of steel
shear dowels to attach the concrete deck. The lay-up of the shell was the same as the
first bending test (lay-up #1) as shown in Table 5-7. A pair of 38mm (1.5 in.) holes
was drilled 102mm (4 in.) apart every 610mm (2 ft) along the shell. A pair of #6 bars
were placed into these holes prior to pumping the shell (see Figure 5-85).
Displacements were monitored at 5 points along the bottom surface of the shell
along with relative displacements between the deck and shell and endblock rotations.
The location of these potentiometers is shown in Figure 5-83. Strain gages were
placed on the shell as shown in Figure 5-84. One dowel of each pair was instrumented
with two strain gages, one on the north side and one on the south side of the bar at the
interface between the shell and the deck as shown in Figure 5-85. The top mat of steel
was gaged to study the strain profile across the section. These gage locations as well
as four displacement devices and one concrete gage are shown in Figure 5-86. Strain
gages were utilized to monitor the effective stress concentration around the
penetrations in the shell for the shear dowels. The gage layout and designation for
these gages is shown in Figure 5-87. Two locations on the top surface of the shell
were gaged, one in the shear span around the set of dowels 1.52m (5 ft) from the north
138
abutment (global location 1) and one in the constant moment region 3.35m (11 ft)
from the north abutment (global location 2).
SL1 SL2 SL3 SL4 SL5 SL6 SL7
D1 D2 D3 D4 D51.22m(4'-0")
1.32m(4'-4") 813mm
(2'-8")
305mm(1'-0")
610mm(2'-0") TYP 1.52m
(5'-0")
SUP-N SUP-S
ROT-N ROT-S
610mm(2'-0")
610mm(2'-0")
813mm(2'-8")
1.32m(4'-4")
Figure 5-83 Displacement Instrumentation for Filled Shell With Slab Test
A
A C
C D
D
A-A
12
34
567
8
C-C
5B-B
B
B
1
5D-D
1.83m(6'-0")
1.37m(4'-6")
1'-6" 1.22m(4'-0")
DESIGNATION
LONGITUDINALLOCATION
A4P
RADIAL LOCATION
ORIENTATIONL - LONGITUDINALH - HOOPP - +45M - -45
Figure 5-84 Strain Gage Locations and Designation for Shell
457mm
139
1 2 3 4 5 6 7 8 9 10 11
N
J3SJOINT
LOCATION
NORTHSOUTH
DESIGNATION
GAGE LOCATION305mm(1'-0") 610mm
(2'-0")TYP
Figure 5-85 Strain Gages Placed on Shear Connection Dowels
1 2 3 4 5 6 7 8 9 10
1
2
3
4
5
LOCATION
BA
R #
610mm(2'-0")TYP
457m
m(1
'-6")
TYP
-STEEL STRAIN GAGE LOCATION-CONCRETE STRAIN GAGE LOCATION
305mm(1'-0")
-VERTICAL DISPLACEMENT TRANSDUCERS* ALL TRANSDUCERS HAVE PREFIX DS (I.E. DSNE)
76.2mm(3") TYP
76m
m(3
") T
YP
SE
SW
NE
NW
SDC
DECK OUTLINE
Figure 5-86 Instrumentation Locations and Designation for Top Mat of Steel Reinforcement in Deck
140
8
9
10
11
12
13
14
15
5
6
7
2
3
4
1
51m
m2�
51m
m2�
76mm3�
76mm3�
76mm3�
SC1_3DESIGNATION
STRESS CONCENTRATIONGLOBAL LOCATION
GAGE LOCATION
Figure 5-87 Stress Concentration Strain Gage Locations and Designation
Initially all instrumentation was zeroed with the specimen supported by
scaffolding. The scaffolding was then removed and a scan was taken to evaluate the
dead load deflections and stresses. The specimen was cycled with actuator loads
chosen to match the shear demand on the connection dowels at peak service conditions
in a prototype bridge. This load was calculated to be approximately 67 kN (15 kips)
per actuator. Three cycles were applied at roughly 6 minutes per cycle to see if any
degradation in stiffness was evident. No degradation was seen after these initial cycles
so 1000 cycles were applied at a frequency of 1 Hz. The load was then increased in
steps of 22 kN (5 kips) per actuator until twice the service load had been applied (133
kN (30 kips) per actuator). At this level 100 cycles were applied to the specimen at 1
Hz. Still no degradation was evident. The load was then increased again in 22.25 kN
141
(5 kip) increments until three times the service load had been achieved (200 kN (45)
kips per actuator). Again at this load level 100 cycles were applied and a slight
softening was observed. The load was then stepped up to 223 kN (50 kips) per
actuator at which time the shear connection between the shell and the slab on the north
side of the specimen began to slip. Significant softening was observed. As the load
was increased to 245 kN (55 kips) per actuator the shear connection on the south side
began to slip and again softening was observed. The load was increased to an ultimate
value of 427 kN (95.9 kips) per actuator at which time the load plateaued due to
excessive deformations in the endblocks.
The load vs. deformation plot is shown in Figure 5-88. For this and the
following plots the cyclic loading has been removed (envelope is shown). Figure 5-89
shows the strain profile across the section for various load levels. Note that plane
sections remain essentially plane until the shear joint between the shell and deck slips
at which time the members begin to bend independently. The shear dowels showed
little to no strains until the slippage occurred at which time some of them yielded
immediately. This shows that the shear was being primarily carried by the friction
between the concrete and the carbon shell. When this friction joint failed the dowels
had insufficient strength to maintain the level of shear necessary to keep the system
acting monolithically. The results from the stress concentration arrays are presented in
Figure 5-90 and Figure 5-91 respectively for the shear and constant moment regions.
These plots show that there is a significant stress concentration around the
penetrations. A parametric study of this effect is presented in Chapter 8.
142
Center Displacement (in)
0 1 2 3 4 5 6
Lo
ad P
er A
ctu
ato
r (k
ips)
0
20
40
60
80
100
Center Displacement (mm)
0 20 40 60 80 100 120 140 160
Lo
ad P
er A
ctu
ato
r (k
N)
0
100
200
300
400
500
Figure 5-88 Load Displacement Envelope for Filled Shell With Slab
Longitudinal Strain (microstrain)
-2000 -1000 0 1000 2000 3000 4000 5000 6000
Dis
tan
ce F
rom
Sec
tio
n B
ott
om
(m
m)
0
100
200
300
400
500
600
Dis
tan
ce F
rom
Sec
tio
n B
ott
om
(in
)
0
5
10
15
200 kN (0 kips)50 kN (11.25 kips)100 kN (22.5 kips)150 kN (33.7 kips)200 kN (44.9 kips)250 kN (56.2 kips)300 kN (67.4 kips)350 kN (78.7 kips)400 kN (89.9 kips)425 kN (95.5 kips)
Top of Shell
Load
25kN=5.62 kips
Figure 5-89 Strain Profile Across Section for Filled Shell With Slab
143
Actuator Load (kN)
0 25 50 75 100 125 150 175 200
Str
ain
(m
icro
-str
ain
)
-50
0
50
100
150
200
250
300
350
400
450
500
550
600
Actuator Load (kips)
0 10 20 30 40
Far Field LongitudinalHoop
Figure 5-90 Stress Concentration Around Penetration in Shear Span
Actuator Load (kN)
0 25 50 75 100 125 150 175 200
Str
ain
(m
icro
-str
ain
)
-50
0
50
100
150
200
250
300
350
400
450
500
550
600
Actuator Load (kips)
0 10 20 30 40
Far FieldLongitudinalHoop
Figure 5-91 Stress Concentration in the Const. Moment region
144
In the following chapter the experimental data presented above is used to
validate the analytical models presented in Chapter 4.
145
6. CORRELATION OF ANALYTICAL MODELS TO EXPERIMENTAL DATA
This chapter presents the correlations between the analytical models put forth
in Chapter 4 and the experimental investigations described in Chapter 5. The
experimental data from the small scale bending tests shows more scatter than from the
large scale tests. It is felt that the geometry of the small scale test specimens made it
difficult to get data unpolluted by the boundary effects as the gage locations were
fairly close to the points of load application and support. The correlations are
presented for the circular sections first, followed by comparisons for the conrec
sections.
6.1 Circular Shells
6.1.1 Small Scale Shells
6.1.1.1 Compression
The analytical model presented in Chapter 4 for the compression behavior of
concrete filled FRP shells is here correlated to the test data from the small scale
compression cylinders. The analytical model described in Section 4.1.1.2 is largely
based on this test data so good correlation is to be expected. Only the thin and thick all
hoop cylinders were able to be modeled due to the experimental difficulties with the
helical cylinders discussed in Chapter 5. The stress strain behavior of the cylinders is
compared in Figure 6-1. The thicker shell shows a significant increase in load
146
carrying and ultimate strain capacity. The expansion behavior is compared in Figure
6-2 where the radial strain is plotted vs. the axial strain for the six cylinders used in
this analysis. The increased strength cited above for the thick shells correlates with the
decreased radial expansion evident in Figure 6-2. It must be noted that the increase in
strength came at the cost of doubling the shell thickness. A point of diminishing
returns is reached with respect to increasing shell thickness where limited strength
gains are realized from the additional material.
Longitudinal Strain
-0.04-0.03-0.02-0.010.00
Str
ess
(ksi
)
-20
-15
-10
-5
0
Str
ess
(MP
a)
-120
-100
-80
-60
-40
-20
0
ExperimentalAnalytical
Thick Circular Shells
Thin Circular Shells
Figure 6-1 Concrete Stress Vs. Strain for Small Scale Compression Specimens
147
Longitudinal Strain
-0.04-0.03-0.02-0.010.00
Ho
op
Str
ain
0.000
0.002
0.004
0.006
0.008ExperimentalAnalytical
Thin Circular Shells
Thick Circular Shells
Figure 6-2 Radial Vs. Longitudinal Strain for Small Scale Comp. Specimens
6.1.1.2 Bending
Two small scale circular bending tests were performed as described in Section
5.1.3.1. The member stiffnesses are compared to the analytical predictions for both of
the circular shells in Figure 6-3. It can be seen that the member stiffnesses are over
predicted by the analytical model. The analytical stiffness of the shell alone (no
concrete fill) is also plotted. The low stiffness of the specimens seen in these plots is
not explained at this time. The author assumes that additional deflection was being
introduced into the system from the support structures. This additional deflection may
have come from the semicircular pivots designed to hold the circular specimens as no
such discrepancy was seen for the conrec specimens. The extreme fiber strains are
shown in Figure 6-4 and Figure 6-5 for the constant moment region and in Figure 6-
148
6 and Figure 6-7 for the shear span of the thin and thick shells respectively. The
correlation of theses strains also points to support deflection as the source of the
additional deflection since the curvature of the specimens was predicted fairly well by
the models especially for the thick shell. The analytical models developed to this point
can not accurately predict the hoop strain increase seen in the shear span. It is thought
that this increased strain is due to the increased cracking due to the shear loads.
Increased tension strains seen in the thin shell in the shear span (Figure 6-7) further
indicate the poor behavior of this specimen reported in Chapter 5. Shear strains were
predicted using the unbonded model described in Section 4.1.3 due to the fact that
these small scale shells had no ribs. A small amount of slippage between the concrete
core and carbon shell was observed during the tests which verifies the assumption that
the core and shell are not deforming as a unit.
Figure 6-8 and Figure 6-9 present the shear strains for the thin and thick shells
respectively. These plots show the analytical predictions with the minimum and
maximum values of vk presented in Section 4.1.3 (0.16-0.29 MPa units (1.9-3.5 psi
units)) used to show the effect of this parameter on the predicted shear strain. The thin
shell begins displaying shear strains at a shear load close to the minimum concrete
cracking load predicted by the analytical model. The rate at which the shear strain in
the shell increases after this point is lower than predicted by the analytical model. This
indicates that the concrete continues to increase the amount of total shear force it is
resisting beyond the shear at initial cracking. At approximately 56 kN (12.5 kips)
shear a small jump is seen in the shear strain in the shell. This indicates a loss of load
149
carried by the core instantaneously that was then shifted to the shell. After this point
the concrete core begins to pick up addition load again as the rate of increase in the
shell is still not as high as would be predicted by the analytical model which assumes
that all shear load past the initial cracking load is taken by the shell. The thick shell
also began to show shear strains in the shell at the minimum concrete cracking load
predicted. The concrete in this case however continued to carry significant additional
loads as the rate of the increase of shear strain in the shell was substantially less than
predicted by the analytical model. As loading progressed the rate of increase of the
shear strain came closer and closer to that predicted by the model. From these tests it
seems that using the proposed shear model with the lower limit for vk gives a
conservative estimate of the shear strain in the shell. The author would however
recommend that for bending applications a ribbed shell be used to ensure an adequate
shear transfer mechanism exists to force the system to deform as a unit as was done
for the full scale tests described later in this section.
150
Center Displacement (in)
0.0 0.5 1.0 1.5
Lo
ad (
kip
s)
0
10
20
30
40
50
60
70
Center Displacement (mm)
0 10 20 30 40 50
Lo
ad (
kN)
0
50
100
150
200
250
300
350Thin Shell AnalyticalThin Shell ExperimentalThin Shell With No ConcreteThick Shell AnalyticalThick Shell ExperimentalThick Shell With No Concrete
Figure 6-3 Load vs. Displacement for Small Scale Circular Sections
Moment (kN-m)
0 10 20 30 40 50
Str
ain
(m
icro
-str
ain
)
-10000
-5000
0
5000
10000
Moment (kip-in)
0 100 200 300 400c1l_expc5l_expc1h_expc5h_expc1l_anac5l_anac1h_anac5h_ana
c1
5
Figure 6-4 Strains in Constant Moment region for Thin Circular Shell
151
Moment (kN-m)
0 10 20 30 40 50 60 70 80 90 100 110 120
Str
ain
(m
icro
-str
ain
)
-10000
-5000
0
5000
10000
Moment (kip-in)
0 100 200 300 400 500 600 700 800 900 1000c1l_expc5l_expc1h_expc5h_expc1l_anac5l_anac1h_anac5h_ana
c1
5
Figure 6-5 Strains in Constant Moment region for Thick Circular Shell
Moment (kN-m)
0 5 10 15 20
Str
ain
(m
icro
-str
ain
)
-3000
-2000
-1000
0
1000
2000
3000
4000
5000
Moment (kip-in)
0 25 50 75 100 125 150 175a1l_expa5l_expa1h_expa5h_expa1l_anaa5l_anaa1h_anaa5h_ana
a1
5
Figure 6-6 Strains in Shear Area for Thin Circular Shell
152
Moment (kN-m)
0 10 20 30 40 50
Str
ain
(m
icro
-str
ain
)
-5000
-4000
-3000
-2000
-1000
0
1000
2000
3000
4000
5000
6000
Moment (kip-in)
0 100 200 300 400a1l_expa5l_expa1h_expa5h_expa1l_anaa5l_anaa1h_anaa5h_ana
a1
5
Figure 6-7 Strains in Shear Span for Thick Circular Shell
Shear (kips)
0.0 2.5 5.0 7.5 10.0 12.5 15.0
Shear (kN)
0 10 20 30 40 50 60 70
Sh
ear
Str
ain
(m
icro
-str
ain
)
0
2000
4000
6000
8000
10000ExperimentalAnalytical vk=.17 S.I. (2 U.S.)
Analytical vk=.29 S.I. (3.5 U.S.)
Gage Locations
Figure 6-8 Shear Strain in Thin Circular Shell
153
Shear (kips)
0 5 10 15 20 25 30
Shear (kN)
0 20 40 60 80 100 120 140
Sh
ear
Str
ain
(m
icro
-str
ain
)
0
2000
4000
6000
8000
10000
12000ExperimentalAnalytical vk=.17 S.I. (2 U.S.)
Analytical vk=.29 S.I. (3.5 U.S.)
Gage Locations
Figure 6-9 Shear Strain in Thick Circular Shell
6.1.2 Full Scale Specimens
Correlations for the full scale bending tests are presented below. The load vs.
displacement behavior is shown in Figure 6-10 for both of the shells tested. The
second shell is substantially softer than the first as would be expected due to the higher
thickness of the first shell and the higher percentage of helical fibers. The ultimate
displacement predicted for both shells is essentially the same since the model predicts
failure based on a ply stress allowable The neutral axis is shifted towards the
compression side of the section since concrete is assumed to take no tension. The
neutral axis location and the curvature determine the longitudinal strain in the extreme
fibers of the shell which in turn determines the peak ply stresses. The softer shell
154
forces the neutral axis more towards the compression side of the member causing a
slight decrease in the allowable curvature and a slight decrease in the ultimate
allowable deflection. Of course as mentioned in Chapter 5 the first ply failure model
does not take into account the compression buckling failure. This buckling failure is
addressed in Chapter 8. The extreme fiber longitudinal and hoop strains in the constant
moment region are shown in Figure 6-11 and in the shear span in Figure 6-12 for
shell #1. These plots are repeated for shell #2 in Figure 6-13 and Figure 6-14. Good
correlation is seen in all areas between the analytical model predictions and the
experimental results. Shell #2 showed increased tension strains at the higher load
levels in both the constant moment and shear spans as seen in Figure 6-13 and Figure
6-14. As stated in Chapter 5 it is believed that the relatively thick layer of pure hoop
fibers on the outside of the shell allowed for increased matrix cracking in the
longitudinal direction which led to the increased tension strains evident in these plots.
The analytical models predict the onset of microcracking for this shell at a moment of
approximately 293 kN-m (2600 kip-in.) which is below the moment at which the
increased strains are seen (600 kN-m (5000 kip-in.) in the constant moment region and
400 kN-m (3500 kip-in.) in the shear span). The shear strains are compared to the
analytical predictions for the two shells in Figure 6-15 and Figure 6-16. Shell #1
showed a gradual increase in the rate that the shell picked up shear strains which
matched the analytical prediction generated with the second model presented in
Section 4.1.3 for systems with an adequate shear transfer mechanism between the shell
155
and the concrete core. The results for shell #2 were very similar with slightly more of
a bilinear nature evident in the response.
Displacement (mm)
0 50 100 150 200 250 300 350 400
Lo
ad P
er A
ctu
ato
r (k
N)
0
50
100
150
200
250
300
350
400
450
500
Center Displacement (in)
0 2 4 6 8 10 12 14
Lo
ad P
er A
ctu
ato
r (k
ips)
0
20
40
60
80
100Shell #1 Exp.Shell #2 Exp.Shell #1 Ana.Shell #2 Ana.
Figure 6-10 Load Displacement Curves for Full Scale Filled Shell Tests
156
Moment (kN-m)
0 100 200 300 400 500 600 700 800
Str
ain
(m
icro
-str
ain
)
-8000
-6000
-4000
-2000
0
2000
4000
6000
8000
Moment (kip-in)
0 1000 2000 3000 4000 5000 6000 7000c1l_expc1h_expc5l_expc5h_expc1l_anac1h_anac5l_anac5h_ana
c1
5
Figure 6-11 Extreme Fiber Strains in Constant Moment region for Shell #1
Moment (kN-m)
0 100 200 300 400 500 600
Str
ain
(m
icro
-str
ain
)
-4000
-2000
0
2000
4000
6000
Moment (kip-in)
0 1000 2000 3000 4000 5000a1l_expa1h_expa5l_expa5h_expa1l_anaa1h_anaa5l_anaa5h_ana
a1
5
Figure 6-12 Extreme Fiber Strains in Shear Span for Shell #1
157
Moment (kN-m)
0 100 200 300 400 500 600 700 800
Str
ain
(m
icro
-str
ain
)
-10000
-7500
-5000
-2500
0
2500
5000
7500
10000
12500
15000
Moment (kip-in)
0 1000 2000 3000 4000 5000 6000 7000c1l_expc1h_expc5l_expc5h_expc1l_anac1h_anac5l_anac5h_ana
c1
5
Figure 6-13 Extreme Fiber Strains in Constant Moment region for Shell #2
Moment (kN-m)
0 100 200 300 400 500 600
Str
ain
(m
icro
-str
ain
)
-6000
-4000
-2000
0
2000
4000
6000
8000
Moment (kip-in)
0 1000 2000 3000 4000 5000a1l_expa1h_expa5l_expa5h_expa1l_anaa1h_anaa5l_anaa5h_ana
a1
5
Figure 6-14 Extreme Fiber Strains in Shear Span for Shell #2
158
Shear (kN)
0 50 100 150 200 250 300
Sh
ear
Str
ain
(m
icro
-str
ain
)
0
500
1000
1500
2000
2500
3000
Shear (kips)
0 10 20 30 40 50 60
ExperimentalAnalytical
Gage Locations
Figure 6-15 Shell Centerline Shear Strains for Shell #1
Shear (kN)
0 50 100 150 200 250 300
Sh
ear
Str
ain
(m
icro
-str
ain
)
0
500
1000
1500
2000
2500
3000
Shear (kips)
0 10 20 30 40 50 60
ExperimentalAnalytical
Gage Locations
Figure 6-16 Shell Centerline Shear Strains for Shell #2
159
6.2 Conrec Shells
6.2.1 Compression
The conrec specimens tested in compression were modeled with finite element
analysis utilizing the concrete models developed for the circular sections as described
in Section 4.2.1. Figure 6-17 shows the load vs. axial strain relation as calculated
from the finite element analysis compared to the experimental results for the thick all
hoop cylinder compression tests described in Section 5.1.2.2. The experimental data
shows a definite kink in the stress strain response of the section that is not picked up
by the analytical approach. It is felt that this kink is due to the rapid expansion of the
concrete core after cracking that occurs due to the poor confinement of the concrete by
the flat sections of the conrec shell. As described in Section 4.1.1.2 the maximum
equivalent tangent Poisson�s ratio for little to no confinement is assumed to be 0.5. As
was described in Chapter 2 an unconfined cylinder or even a cylinder confined with
mild steel can experience volume expansion after the initial loading stages. This
volume expansion implies an equivalent tangent Poisson�s ratio above 0.5. Due to the
lack of data in this range it is not at this time possible to accurately predict the
behavior of the concrete at these low levels of confinement. The finite element
methods used here assume an isotropic elastic material for the concrete which does not
allow for a Poisson�s ratio equal to or greater than 0.5 due to the problems associated
with incompressible material models in finite element calculations. It may be possible
in the future as data becomes available for the lower confinement levels to extend this
method by using a nonisotropic material model in the finite element calculations that
160
would allow for different moduli and Poisson�s ratios to exist for the concrete in
different directions. Such a model could theoretically be introduced into a nonlinear
finite element code. The experimental radial strains in the shell for the center of the
flat , the tangent point of the radius, and the center of the radius are compared to the
analytical model predictions in Figure 6-18. The rapid increase in the hoop strain at
the center of the flat implies that the expansion postulated above is in fact taking
place. As the loading progresses the model and experimental results show the same
trend. It is felt that to accurately predict the behavior of noncircular sections not only a
nonlinear material model must be used but also nonlinear geometry must be
considered. For the conrec section being considered here as the concrete expands the
flat sides bulge out and the section will more closely resemble a circular section as the
loading progresses. This deformed section is better able to resist further expansion
than the original geometry.
161
Longitudinal Strain (micro-strain)
-40000-30000-20000-100000
Lo
ad (
kN)
-3000
-2500
-2000
-1500
-1000
-500
0
Lo
ad (
kip
s)
-600
-500
-400
-300
-200
-100
0
ExperimentalFinite Element
Figure 6-17 Load vs. Longitudinal Strain for Thick Conrec Cylinders
Longitudinal Strain (micro-strain)
-20000-15000-10000-50000
Ho
op
Str
ain
(m
icro
-str
ain
)
0
1000
2000
3000
4000
5000
6000
ExperimentalAnalytical
Figure 6-18 Hoop Strains vs. Longitudinal Strain in Thick Conrec Section
162
6.2.2 Bending
The small scale conrec bending specimens described in Section 5.1.3.2 are
here used to demonstrate the ability of the models presented in Chapter 4 to predict the
behavior of a conrec section in bending. The load vs. displacement curves for both the
thin and thick conrec shells are shown in Figure 6-19 along with the predicted
stiffness of the shells alone with no concrete fill. Good correlation was seen between
the experimental and analytical results with none of the additional displacement seen
for the circular sections evident for the conrecs. The extreme fiber strains in the
longitudinal and hoop directions in the constant moment region for the thin and thick
shells are shown in Figure 6-20 and Figure 6-21 respectively. The same data is shown
for the shear span in Figure 6-22 and Figure 6-23. For the thin shell a lag is seen in
the tension strain at the initiation of loading in both the constant moment and shear
spans and the extreme fiber strains are slightly overpredicted by the model. For the
thick shell good correlation was seen between the experimental and analytical values
of the longitudinal strains. The hoop strains predicted utilizing the �equivalent circular
section� described in Section 4.2.2 are reasonable estimates of the strains seen in the
tests for the constant moment region especially for the thick shell although in the shear
span the increased hoop strains seen in the circular sections are also evident for the
conrecs. Shear strains on the centerline of the shell are shown along with the
predictions generated using the shear model presented in Section 4.1.3 assuming no
shear transfer mechanism is present between the shell and the concrete core in Figure
6-24 and Figure 6-25 for the thin and thick shells respectively. The thin shell begins
163
to show significant shear strains at a shear force level equal to that predicted by the
shear cracking relation presented with the lower limit for vk. The rate that the shell
shear strain increases is lower than the model prediction indicating the continuing
increase in shear carried by the concrete core. At a shear load of approximately 52 kN
(12 kips) the rate of increase of the shear strains increases to a level equivalent to that
predicted which indicates no additional shear is being taken by the concrete core. The
thick conrec section also began to show shear strains in the shell in the range predicted
by the shear cracking model. The shear strains increased at a rate again slower than
predicted up to a shear load of approximately 130 kN (30 kips) at which point a
sudden increase of the shear strain in the shell was seen. This sudden increase
indicates a loss of shear load being carried in the concrete core which could indicate a
loss of aggregate interlock in the shear span common for lightweight concrete as the
shear strength of the aggregate is low compared to standard aggregate. The shear
strains in the shell after this point continued to increase at a rate which indicates no
additional shear being taken by the concrete core.
In the following chapter the analytical models will be used to investigate the
effects of varying the parameters that effect the response of the system.
164
Center Displacement (mm)
0 5 10 15 20 25 30 35
Lo
ad (
kN)
0
50
100
150
200
250
300
350
Center Displacement (in)
0.00 0.25 0.50 0.75 1.00 1.25
Lo
ad (
kip
s)
0
10
20
30
40
50
60
70Thin Shell ExperimentalThick Shell ExperimentalThin Shell AnalyticalThick Shell AnalyticalThin Shell With No ConcreteThick Shell With No Concrete
Figure 6-19 Load Displacement for Conrec Bending Specimens
Moment (kN-m)
0 10 20 30 40 50 60
Str
ain
(m
icro
-str
ain
)
-6000
-4000
-2000
0
2000
4000
6000
8000
10000
Moment (kip-in)
0 100 200 300 400 500c1l_expc1h_expc6l_expc6h_expc1l_anac1h_anac6l_anac6h_ana
1
6
c
Figure 6-20 Strains in Constant Moment region for Thin Conrec Shell
165
Moment (kN-m)
0 25 50 75 100 125 150
Str
ain
(m
icro
-str
ain
)
-10000
-5000
0
5000
10000
15000
Moment (kip-in)
0 200 400 600 800 1000 1200c1l_expc1h_expc6l_expc6h_expc1l_anac1h_anac6l_anac6h_ana
1
6
c
Figure 6-21 Strains in Constant Moment region for Thick Conrec Shell
Moment (kN-m)
0 5 10 15 20 25
Str
ain
(m
icro
-str
ain
)
-3000
-2000
-1000
0
1000
2000
3000
4000
Moment (kip-in)
0 50 100 150 200a1l_expa1h_expa6l_expa6h_expa1l_anaa1h_anaa6l_anaa6h_ana
1
6
c
Figure 6-22 Strains in Shear Area for Thin Conrec Shell
166
Moment (kN-m)
0 10 20 30 40 50 60
Str
ain
(m
icro
-str
ain
)
-4000
-3000
-2000
-1000
0
1000
2000
3000
4000
5000
6000
Moment (kip-in)
0 100 200 300 400 500a1l_expa1h_expa6l_expa6h_expa1l_anaa1h_anaa6l_anaa6h_ana
1
6
c
Figure 6-23 Strains in Shear Area for Thick Conrec Shell
Shear (kN)
0 20 40 60 80
Sh
ear
Str
ain
(m
icro
-str
ain
)
0
2500
5000
7500
10000
12500
15000
Shear (kips)
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5
ExperimentalAnalytical vk=.17 S.I. (2 U.S.)
Analytical vk=.29 S.I. (3.5 U.S.)
Gage Locations
Figure 6-24 Shear Strain in Thin Conrec Shell
167
Shear (kN)
0 25 50 75 100 125 150 175 200
Sh
ear
Str
ain
(m
icro
-str
ain
)
0
2500
5000
7500
10000
12500
15000
17500
20000
Shear (kips)
0 5 10 15 20 25 30 35 40
ExperimentalAnalytical vk=.17 S.I. (2 U.S.)
Analytical vk=.29 S.I. (3.5 U.S.)
Gage Locations
Figure 6-25 Shear Strain in Thick Conrec Shell
168
7. PARAMETER STUDIES OF MATERIAL, LAY-UP, THICKNESS AND SHAPE VARIATIONS
In the following sections the behavior of concrete filled fiber reinforced shells
will be investigated varying a range of parameters. The parameters investigated are
radius to thickness ratios, composite architectures materials and shape variations. Four
radius to thickness ratios are studied varying from 10 to 25 in increments of 5. The
higher the radius to thickness ratio the greater the influence of the concrete core on the
behavior of the system. The composite shell architectures are composed of ±10o plies
and 90o plies. The percentage of helical plies (±10o) is varied from 10% to 90% of the
total lay-up thickness. Carbon and E-Glass composites are used with the pertinent ply
properties listed in Table 7-1. The concrete properties assumed for this analysis are
those of the compression cylinders tested which were used to develop the analytical
model presented in Chapter 4. This is a 45.5 MPa (6.6 ksi) lightweight concrete with a
modulus of 20.7GPa (3.0 msi).
The cost per unit weight of the finished FRP shell is highly dependent on the
cost of the constituent materials and the manufacturing method employed. These
values are constantly in flux. For the materials being investigated in this section at the
time of publication of this dissertation a reasonable estimate for a finished shell
produced with E-glass is $8.00/lb. and for carbon is $25.00/lb. This makes a carbon
shell 3.12 times more expensive per pound than an E-glass shell. The efficacy of these
169
materials in compression and flexure is compared on a weight basis in the following
sections.
Table 7-1 Ply Properties for Parameter Studies
E-Glass Carbon
E1 39 GPa
(5.66 msi)
121 GPa
(17.5 msi)
E2 6.90 GPa
(1.00 msi)
6.90 GPa
(1.00 msi)
ν12 .25 .30
Specific
Gravity
1.8 1.61
7.1 Circular Shells
7.1.1 Compression Behavior
In this section the compression behavior of circular shells will be investigated.
The stress is reported as a normalized ratio of the concrete stress at the allowable
longitudinal strain of the shell, which is taken as 1.2% for carbon and 1.4% for E-
glass, to the ultimate compression stress of the unconfined concrete, f�c. The plots
presented below compare the concrete response for a given radius to thickness ratio
and material for all composite architectures. Figure 7-1 is for carbon and Figure 7-2
is for E-glass.
170
The trends follow intuition with the highest confinement offered by smaller
radius to thickness ratios, more 90o plies and stiffer materials. The most confinement
and thus highest concrete strength is achieved with the predominantly hoop carbon
fiber shells giving a ratio of maximum concrete stress to f�c of 1.85 for a shell with a
radius to flat ratio of 10.
A comparison of the confinement efficiency of the two materials studied in this
section is presented in Figure 7-3. This plot shows the ratio of the weight of an E-
glass shell to the weight of a carbon shell required to give the same peak stress with
the same percentage of hoop and helical plies. This plot demonstrates that at the
current cost estimates for the two material systems carbon is more efficient at
confining the concrete core for all but the shells with a high percentage of helical
fibers (over ~75%).
171
% Helical Plies
10 20 30 40 50 60 70 80 90
σ 1/f' c
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
R/t=10R/t=15R/t=20R/t=25
Figure 7-1 Compression Behavior of Carbon Epoxy Shells
% Helical Plies
10 20 30 40 50 60 70 80 90
σ 1/f' c
1.24
1.26
1.28
1.30
1.32
1.34
1.36
1.38
1.40
1.42
1.44
R/t=10R/t=15R/t=20R/t=25
Figure 7-2 Compression Behavior of E-Glass Shells
172
% Helicals
10 20 30 40 50 60 70 80 90
Wei
gh
t E
-Gla
ss /
Wei
gh
t C
arb
on
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
R/t=10R/t=15R/t=20R/t=25
Equal Cost At Current Cost Estimates
Figure 7-3 Confinement Efficiency of E-Glass vs. Carbon Shells
7.1.2 Bending Behavior
The moment in the section can be written as described in equation 7-1. The
first term on the right side of the equation is the contribution from the shell and the
second term is the contribution from the concrete core. The geometric properties for
this relation are shown in Figure 7-4.
{ } { }[ ]M tR R R c R R R c dLo= − − + − −�2 2 2
1
180σ θ θ σ θ θcos( ) ( ) sin ( ) cos( ) ( ) (7-1)
With the radius to thickness ratio held constant and the lay-up unchanged the neutral
axis position (c) is proportional to the radius. Thus the thickness and neutral axis
173
position can both be expressed as a constant function of the radius as shown in
equation 7-2.
t C Rc C R
==
1
2
(7-2)
Combining equations 7-1 and 7-2 it can be seen that the moment is proportional to the
cube of the radius as shown in equation 7-3.
{ } { }[ ]M R C C C dLo= − − + − −�2 1 13
1 22
1 2
180σ θ θ σ θ θcos( ) ( ) sin ( ) cos( ) ( ) (7-3)
θ
c
N.A.
Rt
Figure 7-4 Geometry for Moment Calculation
The figures below depict the moment curvature response for the range of
radius to thickness ratios, composite architectures and materials described above.
Figure 7-5 through Figure 7-8 are for concrete filled carbon shells and Figure 7-9
through Figure 7-12 are for concrete filled E-glass shells. The moment is normalized
by the inverse of the cube of the radius as explained above and the curvature is
normalized by the radius to generalize the results. For most structural applications
174
these composite members tend to be stiffness critical. To achieve the highest stiffness
it is desirable to place as many fibers close to the longitudinal axis (0o direction) as
possible. With ±10o helical fibers the Poisson�s ratio for loading in the longitudinal
direction of the shell is greater than the initial Poisson�s ratio of the concrete core if
the percentage of helical plies exceeds approximately 85% for carbon epoxy and 80%
for E-glass. This situation should be avoided so the shell does not expand faster than
the concrete core in the initial stages of loading and offer no resistance to cracking.
This analysis does reflect the greater displacement capacity of the fiberglass shells due
to the higher ultimate strain allowable. The peak moment and curvature reported here
correspond to the first ply failure described in Chapter 3. As described earlier these
values are used as theoretical maximum values since they do not take into account
local buckling of the shell.
175
φr
0.000 0.002 0.004 0.006 0.008 0.010 0.012
M/r
3 (M
N-m
/m3 )
0.0e+0
1.0e+5
2.0e+5
3.0e+5
4.0e+5
5.0e+5
M/r
3 (ki
p-i
n/in
3 )
0.0e+0
1.0e+4
2.0e+4
3.0e+4
4.0e+4
5.0e+4
6.0e+4
7.0e+4
10%
90%
% Helicals
Figure 7-5 Normalized Moment Curvature, R/t=10, Carbon Epoxy Shell
φr
0.000 0.002 0.004 0.006 0.008 0.010 0.012
M/r
3 (M
N-m
/m3 )
0.0e+0
5.0e+4
1.0e+5
1.5e+5
2.0e+5
2.5e+5
3.0e+5
3.5e+5
M/r
3 (k
ip-i
n/in
3 )
0.0e+0
1.0e+4
2.0e+4
3.0e+4
4.0e+4
5.0e+4
10%
90%
% Helicals
Figure 7-6 Normalized Moment Curvature, R/t=15, Carbon Epoxy Shell
176
φr
0.000 0.002 0.004 0.006 0.008 0.010 0.012
M/r
3 (MN
-m/m
3 )
0.0e+0
5.0e+4
1.0e+5
1.5e+5
2.0e+5
2.5e+5
M/r
3 (k
ip-i
n/in
3 )
0.0e+0
5.0e+3
1.0e+4
1.5e+4
2.0e+4
2.5e+4
3.0e+4
3.5e+4
10%
90%
% Helicals
Figure 7-7 Normalized Moment Curvature, R/t=20, Carbon Epoxy Shell
φr
0.000 0.002 0.004 0.006 0.008 0.010 0.012
M/r
3 (M
N-m
/m3 )
0.0e+0
5.0e+4
1.0e+5
1.5e+5
2.0e+5
M/r
3 (ki
p-i
n/in
3 )
0.0e+0
5.0e+3
1.0e+4
1.5e+4
2.0e+4
2.5e+4
10%
90%
% Helicals
Figure 7-8 Normalized Moment Curvature, R/t=25, Carbon Epoxy Shell
177
φr
0.000 0.005 0.010 0.015
M/r
3 (M
N-m
/m3 )
0.0e+0
5.0e+4
1.0e+5
1.5e+5
2.0e+5
2.5e+5
M/r
3 (k
ip-i
n/in
3 )
0.0e+0
1.0e+4
2.0e+4
3.0e+4
10%
90%
% Helicals
Figure 7-9 Normalized Moment Curvature, R/t=10, E-Glass Shell
φr
0.000 0.005 0.010 0.015
M/r
3 (M
N-m
/m3 )
0.0e+0
5.0e+4
1.0e+5
1.5e+5
M/r
3 (k
ip-i
n/in
3 )
0.0e+0
5.0e+3
1.0e+4
1.5e+4
2.0e+4
10%
90%% Helicals
Figure 7-10 Normalized Moment Curvature, R/t=15, E-Glass Shell
178
φr
0.000 0.005 0.010 0.015
M/r
3 (M
N-m
/m3 )
0.0e+0
2.0e+4
4.0e+4
6.0e+4
8.0e+4
1.0e+5
1.2e+5
M/r
3 (ki
p-i
n/in
3 )
0.0e+0
5.0e+3
1.0e+4
1.5e+4
10%
90%
% Helicals
Figure 7-11 Normalized Moment Curvature, R/t=20, E-Glass Shell
φr
0.000 0.005 0.010 0.015
M/r
3 (M
N-m
/m3 )
0.0e+0
2.0e+4
4.0e+4
6.0e+4
8.0e+4
1.0e+5
M/r
3 (ki
p-i
n/in
3 )
0.0e+0
2.5e+3
5.0e+3
7.5e+3
1.0e+4
1.3e+4
10%
90%
% Helicals
Figure 7-12 Normalized Moment Curvature, R/t=25, E-Glass Shell
179
The addition of axial load to the section alters the moment curvature response
because the neutral axis is shifted and more of the concrete core is mobilized. This
effect is most notable for large radius to shell thickness ratios where the concrete
makes the greatest contribution to the moment resistance of the section. To
demonstrate this effect moment curvature plots for carbon shells with 10, 50 and 80
percent helical fibers for radius to thickness ratios of 10 and 25 are plotted in Figure
7-13 through Figure 7-18 for axial loads equal to 20, 40 and 60 percent of the capacity
of the unconfined concrete (f�c). These plots demonstrate the increased stiffness
possible with the addition of axial load into the system especially for shells with
predominantly hoop fibers and systems with relatively thin shells where the concrete
offers a greater proportion of the moment resistance. It is envisioned that post
tensioning could be used with this system for certain structural applications to increase
the system stiffness[40].
180
φr
0.000 0.002 0.004 0.006 0.008 0.010
M/r
3 (kN
-m/m
3 )
0e+0
1e+8
2e+8
3e+8
4e+8
5e+8
6e+8
7e+8
8e+8
9e+8
M/r
3 (ki
p-i
n/in
3 )
0.0e+0
2.0e+4
4.0e+4
6.0e+4
8.0e+4
1.0e+5
1.2e+520% f'c40% f'c60% f'c
Axial Load
Figure 7-13 Moment Curvature With Axial Load, R/t=10, 10% Helical Fibers
φr
0.000 0.002 0.004 0.006 0.008 0.010 0.012
M/r
3 (kN
-m/m
3 )
0.0e+0
2.0e+8
4.0e+8
6.0e+8
8.0e+8
1.0e+9
1.2e+9
1.4e+9
1.6e+9
1.8e+9
2.0e+9
M/r
3 (ki
p-i
n/in
3 )
0.0e+0
5.0e+4
1.0e+5
1.5e+5
2.0e+5
2.5e+520% f'c40% f'c60% f'c
Axial Load
Figure 7-14 Moment Curvature With Axial Load, R/t=10, 50% Helical Fibers
181
φr
0.000 0.002 0.004 0.006 0.008 0.010 0.012
M/r
3 (kN
-m/m
3 )
0.0e+0
5.0e+8
1.0e+9
1.5e+9
2.0e+9
2.5e+9
3.0e+9
3.5e+9
M/r
3 (ki
p-i
n/in
3 )
0e+0
1e+5
2e+5
3e+5
4e+5
5e+5
20% f'c40% f'c60% f'c
Axial Load
Figure 7-15 Moment Curvature With Axial Load, R/t=10, 90% Helical Fibers
φr
0.000 0.002 0.004 0.006 0.008 0.010
M/r
3 (
kN-m
/m3 )
0e+0
1e+7
2e+7
3e+7
4e+7
5e+7
6e+7
7e+7
M/r
3 (ki
p-i
n/in
3 )
0
2000
4000
6000
8000
10000
20% f'c40% f'c60% f'c
Axial Load
Figure 7-16 Moment Curvature With Axial Load, R/t=25, 10% Helical Fibers
182
φr
0.000 0.002 0.004 0.006 0.008 0.010
M/r
3 (
kN-m
/m3 )
0.0e+0
2.0e+7
4.0e+7
6.0e+7
8.0e+7
1.0e+8
1.2e+8
M/r
3 (ki
p-i
n/in
3 )
0
4000
8000
12000
1600020% f'c40% f'c60% f'c
Axial Load
Figure 7-17 Moment Curvature With Axial Load, R/t=25, 50% Helical Fibers
φr
0.000 0.002 0.004 0.006 0.008 0.010 0.012
M/r
3 (
kN-m
/m3 )
0e+0
5e+7
1e+8
2e+8
2e+8
M/r
3 (ki
p-i
n/in
3 )
0
5000
10000
15000
20000
2500020% f'c40% f'c60% f'c
Axial Load
Figure 7-18 Moment Curvature With Axial Load, R/t=25, 90% Helical Fibers
183
As with compression the flexural efficiency of the two material systems is
compared on a weight basis. For flexure the comparison is made based on the secant
stiffness of the system. The weight of an E-Glass shell necessary to give the same
secant stiffness of a carbon shell with the same percentage of hoop and helical plies is
shown in Figure 7-19. The secant stiffness for both material systems is evaluated at
the ultimate curvature of the carbon shell. The comparison shown in Figure 7-19
indicates that the E-Glass shell is more efficient for flexure with the current cost
estimates. This does not take into account that E-glass must be isolated from standard
concrete infill (see Section 3.1.1) which could increase the cost of the E-glass system.
% Helicals
10 20 30 40 50 60 70 80 90
Wei
gh
t E
-Gla
ss /
Wei
gh
t C
arb
on
1.5
2.0
2.5
3.0
3.5
R/t=10R/t=15R/t=20R/t=25
Equal Cost At Current Cost Estimates
Figure 7-19 Flexural Stiffness of E-Glass vs. Carbon Shells
184
7.2 Conrec Shells
The motivation to study the conrec sections grew out of the concern that the
circular section is not a very efficient shape for bending applications such as girders or
beams and that connections to a circular section can be difficult due to the line contact
that results if an adjacent member, such as a deck in a beam-and-slab system, is placed
directly on the shell. In this section the moment curvature response of conrec sections
is compared to that of circular sections with the same composite shell architecture.
The comparison is first made based on holding the overall height of the section
constant. As expected a substantial stiffness improvement is realized from reducing
the radius on a conrec section. It must be realized however that the conrec section with
a small radius has a substantial amount more composite material in the shell than does
a circular section of the same overall height. The second comparison in this section
shows the moment curvature response for various shapes if the amount of composite
material is held constant.
For the remainder of this section the section depth (d) is taken to mean the
section total height and the corner radius (r) is taken to mean the radius in the corner
of the conrec section. Figure 7-20 and Figure 7-21 depict the moment curvature
response of three different conrec geometries as compared to a circular section with
the same composite lay-up for two different section depth to thickness ratios (20 and
50). The flat to corner radius ratios are varied from 1/2 to 3 for the conrec sections.
185
The same analysis results are presented in Figure 7-22 for a nominal section
depth to thickness ratio of 20. A nominal ratio is used here since the total height of the
conrec sections is varied to force the total amount of composite material in the shell to
be held constant. Table 7-2 shows the corner radius and flat dimensions used along
with the actual height of the section. All values in the plot are normalized to the same
nominal section depth which is equal to twice the radius of the circular section.
Table 7-2 Geometry of Conrec Sections For Normalized Comparison
Flat/Corner Radius Flat/d Corner Radius/d Height/d
Circle - 0.5 1
1/2 0.1896 0.3792 0.998
1/1 0.3055 0.3055 0.9665
3/1 0.5155 0.1718 0.9095
186
φr
0.000 0.002 0.004 0.006 0.008 0.010 0.012
M/r
3 (ki
p-i
n/in
3 )
0.0e+0
1.0e+4
2.0e+4
3.0e+4
4.0e+4
5.0e+4
6.0e+4
7.0e+4
8.0e+4
9.0e+4
M/r
3 (kN
-m/m
3 )
0.0e+0
1.0e+8
2.0e+8
3.0e+8
4.0e+8
5.0e+8
6.0e+8Circular SectionFlat/Rad=1/2Flat/Rad=1Flat/Rad=3
10%
50%
90%
%Helicals
Figure 7-20 Moment Curvature Response for Conrec Sections, d/t=20
φr
0.000 0.002 0.004 0.006 0.008 0.010 0.012
M/r
3 (ki
p-i
n/in
3 )
0
5000
10000
15000
20000
25000
30000
35000
M/r
3 (kN
-m/m
3 )
0.0e+0
5.0e+7
1.0e+8
1.5e+8
2.0e+8
Circular SectionFlat/Rad=1/2Flat/Rad=1Flat/Rad=3
10%
50%
90%
%Helicals
Figure 7-21 Moment Curvature Response for Conrec Sections, d/t=50
187
φr
0.000 0.002 0.004 0.006 0.008 0.010 0.012
M/r
3 (k
ip-i
n/in
3 )
0
10000
20000
30000
40000
50000
60000
70000
M/r
3 (kN
-m/m
3 )
0.0e+0
5.0e+7
1.0e+8
1.5e+8
2.0e+8
2.5e+8
3.0e+8
3.5e+8
4.0e+8
4.5e+8 Circular SectionFlat/Rad=1/2Flat/Rad=1Flat/Rad=3
10%
50%
90%
%Helicals
Figure 7-22 Normalized Moment Curvature Response for Conrec Sections,
d/t=20
7.3 Hybrid Shells
More than one material may be used to form the fiber reinforced shell. The
ever changing cost benefit equation must be taken into account when selecting the
fiber reinforced composite most suitable for a particular application. As can be seen
from the plots in Section 7.1.1 the response of the confined concrete does not change
much for shells with predominantly helical fibers when we consider carbon or glass
shells. In this section the behavior of a hybrid system with carbon used for the helical
plies and E-glass used for the hoop plies will be investigated.
188
7.3.1 Compression
The compression behavior of a concrete filled fiber reinforced shell is
dominated by the properties of the material used for the hoop direction fibers. There is
seen a slight decrease in the concrete stress due to the fact that the carbon fibers in the
helical plies make the Poisson�s ratio for loading in the longitudinal direction of the
shell slightly higher than for the all glass system thus slightly decreasing the
confinement efficiency of the section. This effect is not more than 3.5% for the
parameter studies presented above.
7.3.2 Bending
Unlike the compression behavior, the bending behavior of the shell is strongly
dominated by the material chosen for the helical plies. As shown in Chapter 6 the
contribution of the concrete core to the total bending stiffness is not great and thus a
change in the confinement of the core does not have a significant effect on the bending
behavior of the system. The moment curvature response of such a hybrid system is
essentially identical to that of a shell wound exclusively with the material used for the
helical plies.
189
8. STRESS CONCENTRATIONS, TENSION STIFFENING AND THERMAL EXPANSION EFFECTS 8.1 STRESS CONCENTRATIONS To use fiber reinforced composite shells as structural members for civil
applications it is necessary to understand the effects of penetrations in the shell that
may be necessary for joining members or attaching nonstructural members. One such
case studied in this document is the penetration used to attach a deck to a composite
shell girder through steel shear dowels. Anisotropic materials can have much higher
stress concentration factors than would be found in an isotropic material. This section
will look at the stress concentration around a circular hole in an orthotropic plate with
and without an elastic inclusion. Finite element models will be used to investigate the
effect of shell curvature on the stress concentrations as well as explore individual ply
stresses. A parametric study of the influence of composite architecture on the stress
concentration factors is performed. In the last section of this chapter a case study is
performed to investigate the stress concentration around the penetrations in the girder
necessary to attach the deck in a beam and slab system.
8.1.1 Closed Form Solution The equations for a circular inclusion in a flat orthotropic infinite plate with
loading along one of the principal directions of the plate were used to parametrically
study the influence of lay-up geometry on the stress concentration factor. The
equivalent stiffness properties for the laminate were determined using classic
190
lamination theory assuming constant strain through the thickness as described in
Chapter 3. This method has been shown to be effective for determining stress
concentrations [41]. The equations presented below are extracted from Lekhnitskii,
Anisotropic Plates [42].
For an orthotropic material the Hooke�s law relations are written as
ε σ σε σ σγ τ
x x y
y x y
xy xy
a aa aa
= += +
=
11 12
12 22
66
(8-1)
The coefficients of deformation can be expressed in terms of the elastic constants as
shown in equations 10-2.
aE
aE E
aE
aG
111
1221
2
12
1
222
6612
1
1
1
=
= − = −
=
=
ν ν
(8-2)
The following relations will be expressed in terms of the coefficients of deformation
for both the infinite plate and the elastic plug. The constants referring to the plug will
be differentiated with a prime. The following constants will be used in this section.
191
k aa
EE
n EE
EG
EE
a aa
aa
= =
= −�
��
�
�� +
= ++
+
22
11
1
2
1
212
1
12
1 4 12 66
11
2 2 22
11
4
2
2
ν
θ θ θ θθ
sin sin cos cos
(8-3)
With
θ − angle around the inclusion measured from the x axis
E1 - plate modulus in the x direction
E2 - plate modulus in the y direction
For the plate and loading as shown in Figure 8-1 the tangential stress around the
inclusion is given by equation 8-4.
x
y
pp θ
Figure 8-1 Infinite Plate With a Circular Inclusion
( )
( )[ ][ ]
σ θ
θ θ
θ θ θ
θθ= −
+ − + + + + − + +
+ − + + + + + + −
p EE
a
n k k n a k a k n a
k k k n a k k a k n a k a
∆∆
∆
∆
11
6
21 2 3
4 2
22 3 4
2 4 24
6
2 1 2 2 1
1 2 2 1
{ sin
( ) ( ) ( )( ) sin cos
( )( ) ( ) ( ) sin cos cos }
(8-4)
192
with
( )[ ]a a a a n k a a a a a kn a a a
a a a a a a a k na a a a n a k a a a a a ka a a a a a a a a k n a a
a a
1 11 11 222
11 11 22 12 122
22 12 12
2 11 11 22 11 12 12
3 11 11 22 22 22 12 12 12 122
4 11 11 22 12 12 12 66 11 12 122
11 22
1
2
= − − + − + − − −
= − + − +
= − + + − + −
= − − + − + + + − −
=
( ) ( ) ( ) ( )
( ) ( ) ( )( )( ) ( ) ( )
( ) ( )[ ( )] ( )(
' ' ' ' '
' '
' ' ' '
' ' ' '
∆ + + + + + − −a a k a a a a a k a a n a a k11 22 22 66 12 11 22 22 11 12 1222' ' ' ' ' ') ( ) ( ) ( )
(8-5)
8.1.2 Parameter Study The equations presented in the previous section have been used to perform a
parametric study of the influence of composite shell architecture on the stress
concentration factor for loading in the longitudinal and hoop directions for both
compression and tension loads Figure 8-2 defines the cases that were studied. Cases 1
and 2 represent loading along the longitudinal direction in tension and compression
respectively, cases 3 and 4 represent loading in the hoop direction in tension and
compression respectively and cases 5 and 6 are for tension in the longitudinal and
hoop direction with no inclusion (open hole). For this analysis the inclusion is
assumed to be the concrete used to fill the shell. Since concrete has very limited
tension capacity an orthotropic plug will be assumed with a very low modulus in the
tension direction. The shell architectures used in this study consist of helical plies at
±100 and hoop plies at 90o and are listed in Table 8-1. The ply properties were
assumed to be those given in Table 5-2.
The results of the parametric analysis are presented in tabular form in Table 8-
2 and graphically in Figure 8-3. The peak stress concentration occurs at 90o from the
193
applied load unless otherwise specified. It can be seen from these results that tension
loading gives stress concentration factors almost identical to the open hole stress
concentration factors in all cases studied. This is due to the fact that the orthotropic
plug was given very low stiffness in the direction of the load for these cases. Further
study also showed that the compression values were almost identical to those
calculated assuming an isotropic plug. This leads to the conclusion that for the values
used in this study the stiffness of the plug transverse to the loading direction does not
have a significant effect on the tangential stress in the composite shell around the hole.
The variation of the stress concentration factor with θ is shown in Figure 8-4 for a
composite lay-up with 80% helical fibers for the first four cases.
194
Case 1 Case 2
Case 3 Case 4
Case 5 Case 6
Figure 8-2 Load Cases for Stress Concentration Study
195
Table 8-1 Composite Architectures for Stress Concentration Study
Lay-up Designation
% Helicals (±10o)
E1 MPa (msi)
E2 MPa (msi)
G12 MPa (msi)
ν12 ν21
1 100 111 (16.1)
7.10 (1.03)
7.86 (1.14)
.703 .0450
2 90 103 (14.9)
18.5 (2.68)
7.58 (1.10)
.258 .0464
3 80 92.4 (13.4)
29.9 (4.33)
7.31 (1.06)
.150 .0485
4 70 82.1 (11.9)
41.3 (5.99)
6.96 (1.01)
.102 .0513
5 60 71.0 (10.3)
52.7 (7.64)
6.67 (.967)
.074 .0549
6 50 60.7 (8.80)
64.1 (9.29)
6.36 (.922)
.056 .0591
7 40 49.9 (7.24)
75.2 (10.9)
6.05 (.878)
.044 .0662
8 30 39.2 (5.68)
86.9 (12.6)
5.74 (.833)
.034 .0754
9 20 28.4 (4.12)
97.9 (14.2)
5.44 (.789)
.027 .0931
10 10 17.7 (2.56)
110 (15.9)
5.13 (.744)
.022 .137
11 0 6.89 (1.0)
121 (17.5)
4.83 (.700)
.017 .300
196
Table 8-2 Stress Concentration Factors
Lay-up
Designation
Case 1 Case 2 Case 3 Case 4 Case 5 Case 6
1 5.73 3.00 -3.6-0o
2.2�55O
1.7�0O
.495
5.52 -3.8�0O
2.2�55O
2 5.24 2.82 2.77 .929 5.20 2.77
3 4.98 2.63 3.25 1.27 4.97 3.25
4 4.79 2.45 3.68 1.57 4.78 3.68
5 4.57 2.24 4.07 1.85 4.57 4.07
6 4.36 2.03 4.45 2.11 4.36 4.45
7 4.11 1.80 4.83 2.35 4.11 4.82
8 3.83 1.54 5.23 2.61 3.83 5.22
9 3.48 1.24 5.64 2.85 3.48 5.62
10 3.03 .896 6.13 3.11 3.03 6.09
11 -3.7� - 0O
2.34
1.6� 0O
.464
6.91 3.39 -4.1� 0O
2.34
6.70
197
% Helical Fibers
0 20 40 60 80 100
Str
ess
Co
nce
ntr
atio
n F
acto
r
0
1
2
3
4
5
6
7Case 1Case 2Case 3Case 4
Figure 8-3 Stress Concentration Factors
Angle From x Axis (Degrees)
0 10 20 30 40 50 60 70 80 90
Str
ess
Co
nce
ntr
atio
n F
acto
r
-3
-2
-1
0
1
2
3
4
5
6
Case 1Case 2Case 3Case 4
Figure 8-4 Tangential Stress Concentration Variation Around Hole for 80%
Helical Shell
198
8.1.3 Case Study A case study was performed on the shear connection used to connect the
concrete deck to the carbon girder in the beam and slab test described in Section 5.2.3
A similar connection using a fiberglass deck system was also investigated. For both of
these structural systems the shear transfer between the deck and the girder is
accomplished through friction, bonding and steel shear dowels. Figure 8-5 shows the
configuration of the connection for the composite deck system which is identical to the
concrete deck system as far as the shear connection to the girder is concerned. The
neutral axis of the concrete deck system is above the top of the girder leading to a state
of biaxial tension in the shell around the penetration. For the system with the
composite deck the neutral axis falls below the top of the girder giving compression in
the longitudinal direction and tension in the hoop direction. Both of these cases will be
evaluated with a finite element model and also using the concentration factors
calculated above.
Polymer Concrete Fiber Glass Deck Panel
Polymer ConcreteSaddle
Light Weight ConcreteSteel Shear Dowel Figure 8-5 Shear Connection
199
A finite element model was utilized to determine the ability of the orthotropic
plate assumption used above to predict the actual stress concentration in an individual
ply of a laminated composite. One quarter of the shell was modeled with symmetry
boundary conditions being used. Quadratic composite plate elements were used so that
the individual ply stresses could be monitored. Orthotropic properties were assumed
for the concrete plug with minimal stiffness assigned to the tension direction to
simulate the inability of the concrete to take tension. The analytical models presented
in Chapter 4 were used to predict the longitudinal and hoop strains present in the top
flange of the shell under maximum service conditions. The longitudinal strain was
applied to the model with imposed nodal displacements along the far field edge. The
reaction of the concrete was replaced with an internal pressure calculated from the
longitudinal and hoop strains. The model used is shown in Figure 8-6. For the case of
the equivalent orthotropic plate superposition is used for the longitudinal and hoop
stresses at the critical location.
The longitudinal strain in the girder at maximum service conditions was found
to be �0.01% for the composite deck system. From the methods described in Chapter 4
the radial strain is determined to be 0.002% and finally the radial pressure is found
from equation 4-6 to be 2.08 kPa (0.30 psi). Applying these loads to the finite element
model the maximum stress around the hole in the ±10o plies was found to be 25.6 MPa
(-3.66 ksi). For the orthotropic plate assumption the far field stresses are obtained from
equations 4-5 and 4-8 giving 9.72 MPa (-1.41 ksi) in the longitudinal Direction and
37.9 kPa (5.5 psi) in the hoop direction. From Figure 8-4 the concentration factor for
200
loading in the longitudinal direction for a shell with 85% ±10o plies is found to be ~2.7
at 90o from the loading direction for case 2 loading. For the hoop loading the
concentration is found to be �1.8 at 00 from the direction of the loading for case 3
(Figure 8-4). Using super-position the total stress around the penetration is found to
be 26.6 MPa (3.85 ksi).
The concrete deck system is evaluated similarly with a longitudinal strain of
0.01%. The pertinent values for both systems are presented in Table 8-3. It can be
seen that the stress concentration predicted in this analysis is substantially greater for
the system with the concrete deck than was recorded in the test. This may be due to
the fact that the gages on the test specimen are not precisely on the edge of the hole
and the concentration dies off quickly. Also the tension strains were so low on the top
surface of the shell that the concrete cracking strain was never reached. This may
allow load to transfer through the concrete and lower the observed stress
concentration.
Table 8-3 Stress Around Penetration for Beam and Slab Shear Connection
Composite Deck Concrete Deck εL -0.01% 0.01% εH 0.002% -0.0005% 8.1.3.1 Radial Pressure -2.08 kPa (-.30 psi) -18.6 kPa (-2.70 psi) F.E.M. Maximum Stress ±100 Plies
-25.6 MPa (-3.71 ksi) 44.5 MPa (6.45 ksi)
Orthotropic Plate Longitudinal Stress
-9.72 MPa (-1.41 ksi) 9.78 MPa (1.42 ksi)
Orthotropic Plate Hoop Stress
37.9 kPa (5.5 psi) 339 kPa (49.2 psi)
Orthotropic Plate Maximum Stress
-26.6 MPa (-3.86 ksi) 48.3 MPa (7.00 ksi)
201
Figure 8-6 Finite Element Model for Stress Concentration Studies
8.2 EFFECTS OF TENSION STIFFENING The relations presented in Section 3.2.2.1 will be used in this section to study
the significance of the tension stiffening effect on the stiffness of a concrete filled
fiber reinforced shell. The first portion of the analysis presents the load versus
deformation curves for a concrete filled fiber reinforced shell in tension. The tension
stiffening relations were then integrated into a moment curvature analysis to
investigate the effect on the bending stiffness. Figure 8-7 shows the stress versus
strain curves for various concrete strengths based on equation (3-27). The light-weight
reduction factor of 0.75 is used to reduce the rupture stress as described in Section
3.2.2.1. This plot demonstrates that the effect of the concrete compressive strength on
the average tension stress is fairly small. For the remainder of this analysis a concrete
202
strength of 27.6 MPa (4 ksi) will be assumed. Figure 8-8 shows the effect of tension
stiffening on a concrete filled carbon shell in pure tension. The geometry of the shell
was held constant with a radius of 171.5mm (6.75 in.) and a shell thickness of
9.65mm (0.38 in.). The composite architecture was assumed to be composed of helical
fibers at ±10o from the longitudinal axis and fibers at 90o from the longitudinal axis
(hoop direction). The percentage of helical fibers was varied from 0% to 100%. In
reinforced concrete analysis the effect of tension stiffening is usually limited to an
effective area of the concrete that is judged to be close enough to a reinforcing bar to
have a stiffening effect. For this analysis the entire concrete section was considered to
give an upper bound to the tension stiffening effect. The curves in Figure 8-8 indicate
that at service conditions the effect of tension stiffening is small. If the ultimate strain
in the composite shell is assumed to be close to 1% then the service strain should not
be over 0.3% which is the maximum strain plotted in Figure 8-8. When analyzed in
bending no significant difference is discernible in the member secant stiffness.
203
Average Strain
0.000 0.002 0.004 0.006 0.008
Ave
rag
e S
tres
s (M
Pa)
0.0
0.5
1.0
1.5
2.0
2.5
Ave
rag
e S
tres
s (k
si)
0
50
100
150
200
250
300
350
f'c=27.6 MPa (4 ksi)f'c=34.5 MPa (5 ksi)f'c=41.4 MPa (6 ksi)
Figure 8-7 Average Stress vs. Average Strain For Tension Stiffening
Axial Strain
0.000 0.001 0.002 0.003 0.004
Lo
ad (
kip
s)
0
100
200
300
400
500
600
700
800
900
Lo
ad (
kN)
0
500
1000
1500
2000
2500
3000
3500
4000
Without Stiffening EffectsWith Stffening Effects
%Helicals
10%
90%
Figure 8-8 Load in Concrete Filled Carbon Shell in Pure Tension With and
Without Tension Stiffening Effects
204
8.3 THERMAL EFFECTS
8.3.1 Thermal Strains in Circular Sections
As shown in Section 3.1.6 coefficients of thermal expansion for the shell in the
longitudinal and hoop direction can be calculated from the composite architecture and
individual ply properties. Once these coefficients are known the stresses induced in the
shell, due to the difference in thermal expansion of the concrete and the composite,
can be determined.
Assuming that the longitudinal stresses are negligible the radial strain in the
concrete is given by
ε ε σ νr rT r
ccE
= − −( )1 . (8-6)
For the composite shell the hoop strain is obtained from
ε ε σH H
T H
HE= + . (8-7)
Using the kinematic and equilibrium relations of equations 4-5 along with 8-6 and 8-7
the radial stress in the concrete can be determined.
σ α ανr
c H
c
c H
T
ER
tE
=−
−+
�
��
�
��
∆ ( )1
(8-8)
The total hoop strain in the shell is then given as
205
( )εα α
ναH
c H
H c
c
H
TtE
RE
T=−
−+
�
��
�
��
+∆
∆( )1
1. (8-9)
The first term on the right hand side of equation 4-26 is the mechanical strain or the
strain that produces stress in the shell and core.
8.3.2 Thermal Testing
To quantify the effects of thermal loading on the concrete filled fiber
reinforced shells a series of experimental investigations was undertaken. These
experiments involved the thermal loading of concrete cylinders wrapped with carbon
fiber. The cylinders were tested in an environmental chamber that allowed free
expansion of the specimens. Three concrete cylinders were wrapped with all hoop
carbon shells by a hand lay-up process. Table 8-4 gives the pertinent ply properties
obtained for this process. The cylinders had one, two and three layers of carbon
respectively. A hollow cylinder for each lay-up was manufactured to be used as a
reference during testing. The hoop strains in the concrete filled shell are compared to
the hoop strains recorded in the reference hollow shell to account for strain variations
in the gage due to temperature changes. Figure 8-9 plots the mechanical hoop strain in
the concrete filled shell for the three different lay-ups tested along with the theoretical
predictions from equation 8-9. It can be seen from this figure that the mechanical
strains are overpredicted by the theoretical models. This could be due to the
coefficients of thermal expansion for the composite or the concrete as these values
were not experimentally determined but nominal values were used.
206
Table 8-4 Ply Properties for Thermal Testing
Lay-Up Property. S.I. U.S.
1-Layer E1 67.4 GPa 9.78 msi
t 0.57mm 0.0225 in.
2-Layer Ε1 75.5 GPa 10.95 msi
t 0.99mm 0.039 in.
3-Layer Ε1 81.2 GPa 11.78 msi
t 1.40mm 0.055 in.
E*t (kN/mm)30 40 50 60 70 80 90 100 110 120
Mec
hani
cal S
train
in H
oop
Dire
ctio
n (m
icro
-stra
in)
0
1
2
3
4
5
6
7
8
9
10
11
E*t (lb*106/in)0.2 0.3 0.4 0.5 0.6
ExperimentalAnalytical
Figure 8-9 Thermally Induced Mechanical Strains Per Degree Centigrade
207
8.3.3 Parameter Studies for Thermally Induced Strains
The stresses induced in the fiber reinforced shell due to the difference in the
coefficient of thermal expansion of the shell and the concrete core will vary depending
on the lay-up chosen, the radius to thickness ratio, and the materials in the shell. The
analytical methods outlined in Sections 3.1.6 and 8.3.1 are here used to investigate
these effects. For this analysis the ply coefficients of thermal expansion are taken to be
those given in Table 3-4 for E-glass-epoxy and carbon-epoxy unidirectional lamina.
The coefficients of thermal expansion for the shell in the hoop and longitudinal
direction are given in graphical form in Figure 8-10. The induced stresses in the shell
are reported for a 55o C (100 oF) temperature increase in Figure 8-11. These figures
indicate that for an E-glass shell with greater than 60% of the fibers in the helical
direction the shell will expand faster than the concrete core. For carbon this does not
occur until the shell has 86% helical fibers. The lower stresses generated in the E-glass
shell occur because the coefficient of thermal expansion is closer to that of the
concrete and the hoop modulus is lower than is the case for the carbon shell.
208
%Helical Fibers
20 40 60 80
Co
effi
cien
t o
f T
her
mal
Exp
ansi
on
(x1
0-6/o
C)
0
2
4
6
8
10
12
14
16
18Carbon LongitudinalCarbon HoopE-Glass LongitudinalE-Glass HoopConcrete
Figure 8-10 Coefficients of Thermal Expansion for Fiber Reinforced Shells
% Helical Plys
20 40 60 80
% o
f A
llow
able
Str
ess
in H
oo
p P
lys
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0Carbon r/t=10Carbon r/t=15Carbon r/t=20Carbon r/t=25E-Glass r/t=10E-Glass r/t=15E-Glass r/t=20E-Glass r/t=25
Figure 8-11 Hoop Stress in Shell 90o Plies Due to a Temperature Rise of 55 oC
209
8.4 Local Compression Buckling of Concrete Filled FRP Shells
The compression buckling problem for concrete filled FRP shells is very
complex due to the anisotropic nature of the materials, the variation in the bending
stiffness as a function of the shell architecture and the geometric stiffness effects
introduced from the expanding concrete core which develops hoop tension in the shell.
Whitney presents approximate solution techniques for thin simply supported
cylindrical shells under compression and internal pressure loading [43]. These
techniques are limited to lay-ups with no extension twist coupling
(A16=A26=B16=B26=0). The radius to thickness ratios considered by Whitney are
several orders of magnitude greater than those considered in this dissertation. The
solution techniques for thin shells neglect the transverse shear deformations of the
shell. For the materials being investigated here the transverse shear stiffness is
generally very low compared to the axial stiffness which can make the shear
deformations much more of an issue in these advanced composite shells than is the
case for metallic shells. Higher order theories which consider the shear deformation of
the shell have been proposed for various boundary conditions [44][45].
The ultimate compression strain seen in the six bending test specimens is
plotted in Figure 8-12 vs. the ratio of the diameter to the thickness. A general increase
in the ultimate strain is seen with increasing thickness with the exception of the first
full scale specimen tested which failed at an ultimate strain level similar to the thin
small scale specimens.
210
D/t
30 35 40 45 50 55 60 65 70
Ult
imat
e C
om
pre
ssio
n S
trai
n (
%)
-1.1
-1.0
-0.9
-0.8
-0.7
-0.6
-0.5
Thin Small ScaleThick Small ScaleFull Scale #1Full Scale #2Thin ConrecThick Conrec
Figure 8-12 Ultimate Buckling Strain for All Bending Specimens
211
9. CONCLUSIONS
The experimental and analytical program outlined in this document have
answered some important questions regarding the use of concrete filled FRP shells for
structural members in civil engineering applications and have opened several new
questions that need to be addressed further.
The analytical modeling of the compression behavior is based on fairly scant
data at this point but the procedure outlined is capable of incorporating new data as it
becomes available for different levels of confinement and for different concrete mixes.
The problems encountered in the compression testing of the cylinders with 85%
helical fibers could be avoided if no end capping was done and instead the cylinders
were machined with two flat parallel ends and tested directly on the steel platens of the
testing machine. An aspect ratio of at least three between the height and diameter of
the cylinders is suggested to minimize the effects of the constraint on the ends. In this
way a database of compression behavior could be built and the relations described in
Chapter 4 for the concrete tangent modulus and equivalent tangent Poisson�s ratio
could be better determined as a function of the concrete strength and weight and
composite architecture.
The bending behavior of these members was modeled fairly well with the
analytical models presented. It was found that the stiffness of the shell in the axial
direction is far and away the controlling factor on the behavior of the system. The
hoop strains in a section under constant moment were predicted well by the models
212
and were found to be small in comparison to the longitudinal strains. The interaction
between the stresses was found to have little effect on the ultimate strength of the
members and an allowable strain in the principal loading direction based on the
composite architecture used is felt to be a justifiable criteria for design purposes. All
specimens tested in bending failed on the compression side of the shell at an ultimate
strain lower than that predicted by a first-ply-failure model even though the highest
strains were on the tension side. This is not surprising as this model does not consider
the stability of the shell. The second large scale filled shell bending test was
successfully designed with more hoop plies on the outside to mitigate this buckling
failure. No attempt was made in the current work to correlate the buckling of the shell
to the thickness and composite architecture but this is felt to be an important area of
study that should be undertaken.
The shear behavior of these members was observed and predictions based on
simple mechanics models were proposed to estimate the shear strain that could be
expected in the fiber reinforced shell. By design no shear failures were observed in the
testing that was completed thus far as the tests were planned to induce bending failure
before shear failure could occur. The ability of the models presented to predict the
shear strain in the shell was fairly good but the additional hoop strain seen especially
along the centerline of the shell is not at present accounted for. A comprehensive
program to quantify the behavior of various composite architectures under shear
loading should be undertaken along with more rigorous modeling to understand the
expansion of the cracked concrete which is driving the hoop strains in the shell up.
213
Concerns over possible induced stresses from thermal expansion effects were
found to be unwarranted for the lay-ups and radius to thickness ratios investigated in
this document. The temperature ranges encountered by a civil structure are not
excessive and the coefficients of thermal expansion for a laminated composite are
generally not as extreme as they can be for an individual ply due to the fact that the
fibers are not all aligned in the same direction. The study here concentrated on the
induced stress in the hoop direction due to a temperature change. The induced stresses
and deformations of a particular structural system assembled with these members must
be studied carefully to ensure that the structure does not encounter any adverse effects
from differential expansion or contraction. Such an analysis is being carried out for the
beam-and-slab assembly currently being tested.
Stress concentration effects will become an important consideration as the
potential uses for these and other composite members in civil applications increase.
The work presented here was intended to give designers a feel for how these
concentration factors change with composite architecture so that if penetrations are
needed appropriate consideration will be given.
The choice of materials for this system must incorporate many different
considerations. Research is underway at many universities and in industry to
characterize the long range environmental durability of composite materials for use in
civil infrastructure applications. From the work presented here it can be seen that with
the current cost estimates the carbon shells are more efficient for providing
confinement and E-glass shells are more efficient for providing bending stiffness. If
214
the cost ratio for carbon/E-glass is reduced to 2.5 carbon becomes the more
economical choice for both applications. The use of hybrid shells to optimize cost is
possible. It has been demonstrated in this dissertation that the compression behavior is
strongly dominated by the hoop fibers and the bending behavior is strongly dominated
by the helical fibers. The fluctuating cost ratio of the various materials must be
considered along with the potential complications in the manufacturing processes
when considering a hybrid shell.
The two shapes investigated in this document show strengths and weaknesses
for different applications. The circular shell is excellent for confining the concrete
core. It was also shown to be a good choice for bending applications if the total height
of the section is not a consideration as it offers superior stiffness for the same amount
of material as compared to the conrec section. If the total height of the section is a
critical consideration substantial stiffness increases are possible with a conrec section.
The loss of confinement efficiency in the conrec section was shown to have little
effect on the overall performance of the system in bending applications. Connections
to adjacent members may be simplified with the use of a conrec section as it offers flat
areas avoiding line contacts.
The construction procedure used to build the test specimens was extended to a
full scale bridge section currently undergoing fatigue testing in the structures lab at
UCSD. The construction procedure was found to be straight forward and the potential
for reduced erection times is felt to be substantial.
236
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